I would like a reference for the result here. Having that $t$ there makes me happy. I would prefer not to have to, in my paper, run through and (not trivially but not too greatly) alter the proof of the standard Weyl inequality to get the result with the $t$, but rather just cite a book or paper. Feel free to just comment so I can delete the question (and edit the Wikipedia page).

$\begingroup$ I think you don't have to go through the proof in detail (in your paper), just indicate the necessary changes relative to a standard reference (e.g. Vaughan's book). Good question, by the way! $\endgroup$ – GH from MO Dec 21 '19 at 21:04

$\begingroup$ I found a PDF reference, but it lacks a detailed proof: math.ucsd.edu/~fan/reading/ross/sums.pdf $\endgroup$ – GH from MO Dec 21 '19 at 21:09

1$\begingroup$ @GHfromMO thank you for your responses. My question wasn't too great, since I kind of misled by sort of implying you can just go through the proof, merely adding $t$'s in the appropriate places. I think some slightly nontrivial (though certainly not difficult) changes need to be made to the proof. I just edited the OP question. Does this new information change your mind re your first comment? $\endgroup$ – mathworker21 Dec 22 '19 at 1:17

1$\begingroup$ If the changes take more than a few lines to explain, then I agree that a precise detailed reference would be more desirable. Let's hope someone knows such a reference! $\endgroup$ – GH from MO Dec 22 '19 at 3:11

$\begingroup$ I don't have it with me to check, but try Montgomery's Ten Lectures on the Interface Between Number Theory and Harmonic Analysis. $\endgroup$ – Greg Martin Dec 22 '19 at 7:39
This form of Weyl's inequality is due to Ivan Matveevich Vinogradov and the relevant reference is the 1927 paper [3]. Precisely, Lemma III at pages 568569 states the following equivalent form: if $$ S=\sum_{x=N+1}^{N+P} e^{2\pi i f(x)},\quad f(x)=\lambda x^n+\ldots+\lambda_n,\label{WS}\tag{WS} $$ and $$ \left\lambda \frac{a}{q}\right<\frac{\tau}{q^2},\quad (a,q)=1, \quad 0<q \le P^n,\quad 1<\tau\le q, $$ then we have $$ S=O\big(P^{1+\epsilon}(1+qP^{n+1})^\sigma(\tau q^{1}+P^{1})^\sigma\big),\quad\sigma=2^{n+1}.\label{WI}\tag{WI} $$
Notes
In the monograph on arithmetical functions by Chandrasekharan [1], Weyl's inequality is developed and proved in a different form, similar to formula (1) in lemma II of [3] (p. 568), which differs from the one given in the Wikipedia entry. In the historical notes ([1], p. 84), Chandrasekharan cites the original works [7] and [8] by Weyl, a preceding note of Hardy and Littlewood and finally refers to the monumental work of Edmund Landau ([2], II, pp. 3146) for "a comprensive formulation". The work of Landau is also cited by Vinogradov ([3] p. 568, footnotes * and **), regarding lemmas I and II.
Since I was not able to find a reference in my trusted source [1], I had a look at the translation of the second edition of the the important monograph [4] included in Vinogradov's "Selected works" [5]: Weyl's inequality wikipedia style is shown as formula (3) ([5], Introduction, p. 185: see also [6], p. 6 formula (5)) of the introduction, but no reference on its origin is stated. Then I decided to have a look at [4] (Introduction, p. 4, formula (4)) and I found the reference right there, just above the following equivalent form of \eqref{WI}: $$ S\le P\gamma $$ where $$ \gamma \ll P^\epsilon\big(P^{1}+tq^{1} + tP^{n+1} + q P^{n}\Big)^\rho \quad \rho =\frac{1}{2^{n1}} $$ and with obvious meaning of $P, q$ and $t$.
In the references [1], [2], [3], [4] and [5] in "Bibliography" section below, the Weyl's sum to be estimated, the summation index set is the same as in formula \eqref{WS} or, equivalently $$ S=\sum_{x=N\color{red}{+1}}^{N+P} e^{2\pi i f(x)}\:\:\text{ or }\:\:S=\sum_{x=N}^{N+P\color{red}{1}} e^{2\pi i f(x)}. $$ The Wikipedia version is the following: $$ S=\sum_{x=M}^{N+M}e^{2\pi if(x)}, $$ and this possibly is the evidence of a typo. However, as noted by GH from MO in his comment, omitting a term only increases the implicit constant in the big $O$ estimate.
As noted by mathworker21, since the estimate $$ S\le P $$ holds trivially and the left side of \eqref{WI} is trivially larger $P$ for $\tau > q$ we can say that this asymptotic estimate holds irrespectively of any upper bound on the value of $\tau$: of course, in such condition it loses its usefulness, since it is far worser than the trivial estimate.
A drawback of formula \eqref{WI} was noted by Vinogradov ([5], pp. 185186, or [6], p. 6): the estimate become quickly less accurate as $n$ increases, since its left side is (far, as he says) larger than $P^{1\sigma}$, and this term tends rapidly to $P$.
Addendum: incidentally I recently noted the work [A1]. The Author, while proving a refinement of \eqref{WI} valid for polynomials $f(x)$ for which the coefficient of the $(n1)$th power is $0$, acknowledges the work of Vinogradov on this formula without citing [3] ([A1] p. 1) and cites Vaughan's monograph as a reference for a proof ([A2] §2.1, lemma 2.4, pp. 1112). This monograph can thus be used as a modern reference on Vinogradov's form of Weyl's inequality for the English reader.
Addendum Bibliography
[A1] Allakov, Ismail A., On an estimate by Weyl and Vinogradov, Sibirskiĭ Matematicheskiĭ Zhurnal 43, No. 1, 913 (2002); translation in Siberian Mathematical Journal 43, No. 1, 14 (2002), MR1888113 ZBL1008.11031.
[A2] Vaughan, Robert C., The HardyLittlewood method, Cambridge Tracts in Mathematics, 125. Cambridge: Cambridge University Press. pp. vii+232 (1997), ISBN: 0521573475, MR1435742 ZBL0868.11046.
Bibliography
[1] Chandrasekharan, Komaravolu, Arithmetical functions, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen. 167. BerlinHeidelbergNew York: SpringerVerlag. XI, 231 p. (1970), MR0277490, ZBL0217.31602.
[2] Landau, Edmund, Vorlesungen über Zahlentheorie. I: Aus der elementaren und additiven Zahlentheorie. II: Aus der analytischen und geometrischen Zahlentheorie. III: Aus der algebraischen Zahlentheorie und über die Fermatsche Vermutung, Leipzig, S. Hirzel. I: xii, 360 S. II: viii, 308 S. III: viii, 342 S. (1927). JFM 53.0123.17.
[3] Vinogradov, Ivan Matveevich, "Démonstration analytique d'un théorème sur la distribution des parties fractionnaires d'un polynôme entier", Bulletin de l’Académie des Sciences de l’Union des Républiques Soviétiques Socialistes, (6) 21, 567578 (1927), JFM 53.0160.02.
[4] Vinogradov, Ivan Matveevich,The method of trigonometrical sums in the theory of numbers. Translated, revised and annotated by K. F. Roth and Anne Davenport, New York: Interscience Publishers Inc. X, 180 p. (1954), MR0062183, ZBL0055.27504.
[5] Vinogradov, Ivan Matveevich, Selected works. Prepared by the Steklov Mathematical Institute of the Academy of Sciences of the USSR on the occasion of his ninetieth birthday. Ed. by L. D. Faddeev, R. V. Gamkrelidze, A. A. Karatsuba, K. K. Mardzhanishvili and E. F. Mishchenko, BerlinHeidelbergNew York: SpringerVerlag pp. viii+401 (1985), ISBN: 3540127887, MR0807530, ZBL0577.01049.
[6] Vinogradov, Ivan Matveevich; Karatsuba, Anatoliĭ Alekseevich, "The method of trigonometric sums in number theory", Proceedings of the Steklov Institute of Mathematics 168, 330 (1986), MR0755892, ZBL0603.10037.
[7] Weyl, Hermann, "Über die Gleichverteilung von Zahlen mod. Eins", Mathematische Annalen 77, 313352 (1916). ZBL46.0278.06.
[8] Weyl, Hermann, "Zur Abschätzung von $\zeta(1+ti)$", Mathematische Zeitschrift 10, 88101 (1921). ZBL48.0346.01.

1$\begingroup$ Thanks a lot for the comprehensive answer. Small/trivial comment: the condition $\tau \le q$ is not really necessary, since the big O term is bigger than $P$ in that case; just wanted to point that out, since Wikipedia just says $\tau \ge 1$. Also, feel free to edit the Wikipedia page if you want (I've never edited one before so might take me more time). $\endgroup$ – mathworker21 Dec 22 '19 at 16:28

1$\begingroup$ What a nice answer! Also, it does not matter whether $S$ is a sum of $P1$ or $P$ terms, since changing one to the other only influences the implied constant in $S=O(\dots)$. (BTW I looked at [4] but somehow I missed this version of Weyl's inequality. Now I see it, in my Dover Publication edition it is (4) on Page 4.) $\endgroup$ – GH from MO Dec 22 '19 at 16:30

1$\begingroup$ @GHfromMO exactly: after you observation, I précised the exact page and form in the relevant note. $\endgroup$ – Daniele Tampieri Dec 22 '19 at 17:07

1$\begingroup$ @mathworker21 I just noticed that I forgot the imaginary exponent in all formulas here!! Thank you so much!!! Now your comment seems obvious to me, since $$ S=\left\sum_{x=N+1}^{N+P} e^{2\pi i f(x)}\right\le \sum_{x=N+1}^{N+P} 1 =P$$ trivially!! I'll correct all the answer and add your simple yet smart comment. Again thank you! $\endgroup$ – Daniele Tampieri Dec 26 '19 at 15:19

1$\begingroup$ @DanieleTampieri thanks for taking the time from your other matters. all makes more sense now. (and yes, all of this is to measure the equidistribution of polynomials mod 1). $\endgroup$ – mathworker21 Dec 26 '19 at 16:58