Morally, any first attack on Waring's problem (e.g. Chapter 2 of Vaughan) works here, but to be rigorous one might modify the argument on the major arcs. If you want to quote an explicit result I suggest Birch; Theorem 1 in Section 7 needs only cosmetic changes to yield the following result:

Theorem (Birch) Let $f(x)\in\mathbb{Z}[x_1,\dotsc,x_k]$ have degree $d$, let $g(x)$ be the leading part. Suppose the singular locus of $g$ has codimension $>(d-1)2^d$ in $\mathbb{C}^k$. Let $\nu\in \mathbb{N}$ and let $\mathscr{B}\subset [-1,1]^k$ be a box of side at most $1$. For some $\delta>0$, the number of integer points $x\in P\mathscr{B}$ with $f(x)=\nu$ is $$ C_{n,f,\mathscr{B}}P^{k-d}+O_{f}(P^{k-d-\delta}) $$ If for each prime $p$ there is $x$ with $f({x})\equiv\nu$ and $\nabla f(x)\not\equiv 0$ mod $p$, and there is a solution $\bar{x}\in \mathscr{B}$ to $g(\bar{x})=1$ with distance $\geq r$ from the real singular locus of $g(\bar{x})=1$, then $C_{n,f,\mathscr{B}}\gg_{f,r} 1$.

Take $\mathscr{B}=\frac{1}{2}\big[(\frac{1}{2s})^{1/d},(\frac{2}{s})^{1/d}\big]^k$ and $P= 2\nu^{1/d}$. If $\nu\gg_f 1$, $k>(d-1)2^d$ and $\sum P_i(x_i)\equiv\nu$ has nonsingular solutions mod $p$ then we deduce that $\sum P_i(x_i)=\nu$ has many solutions in your range.

Why does this follow from Birch's work?

That paper is about some degree $d$ forms $f_1,\dotsc,f_R$. But actually, after the introduction, we can take the $f_i$ to be general degree $d$ polynomials if we replace $f_i$ by its leading part in the following places: the definition of $f^{(i)}_{j_0,\dotsc,j_{d-1}}$ at the start of Section 2; the definition $V(\mu)$ and $V^*$ in formulae (2) and (3) of Section 3; the definition of $I(\mathscr{C};\gamma)$ in formula (9) of Section 5; and throughout Section 6.

In Section 7, Theorem 1 we need to replace $f_i$ by its leading part in the definition of $V^*$ and in the expression $\Phi[f(\bar{x})]$. The theorem above is exactly this in the case $R=1$ of a single polynomial.

The argument really does not change. A little thought to see that Lemma 5.1 is still true, that's it.

A hardcore version of the theorem above would be uniform in the lower degree parts of $f$.

Theorem? Let $f$, $g$, $P$ and $\mathscr{B}$ be as above. Suppose the polynomial $P^{-d} f(P x)$ has coefficients bounded by some fixed constant $C$. If the singular locus of $g$ has codimension $>(d-1)2^d$ in $\mathbb{C}^k$, then the number of integer solutions to $f(x)=0$ in $P\mathscr{B}$ is $$ C_{n,f,\mathscr{B}}P^{k-d}+O_{g,C}(P^{k-d-\delta}) $$ If there are nonsingular solutions mod $p$, and there is a solution $\bar{x}\in \mathscr{B}$ to $P^{-d} f(P \bar{x})=0$ with distance $\geq r$ from the real singular locus of $P^{-d} f(P \bar{x})=0$, then $C_{n,f,\mathscr{B}}\gg_{g,r,C} 1$.

The condition on $P^{-d} f(P \bar{x})=0$ just means that if $x\in P\mathscr{D}$ then $f(x)$ is not much larger than the leading part $g(x)$, so that the lower degree parts do not dominate too badly. The alleged proof is the same as the theorem above, but you push on through sections 5 and 6 of Birch's paper with $f$ remaining inhomogeneous. I have not checked this. Modern technology might give a neater proof.

I should also mention that everything here would work for systems of several forms, with appropriate changes in the condition on the codimension of the singular locus.

Morally, any first attack on Waring's problem (e.g. Chapter 2 of Vaughan) works here, but to be rigorous one might modify the argument on the major arcs. If you want to quote an explicit result I suggest Birch; Theorem 1 in Section 7 needs only cosmetic changes to yield the following result:

Theorem (Birch) Let $f(x)\in\mathbb{Z}[x_1,\dotsc,x_k]$ have degree $d$, let $g(x)$ be the leading part. Suppose the singular locus of $g$ has codimension $>(d-1)2^d$ in $\mathbb{C}^k$. Let $\nu\in \mathbb{N}$ and let $\mathscr{B}\subset [-1,1]^k$ be a box of side at most $1$. For some $\delta>0$, the number of integer points $x\in P\mathscr{B}$ with $f(x)=\nu$ is $$ C_{n,f,\mathscr{B}}P^{k-d}+O_{f}(P^{k-d-\delta}) $$ If for each prime $p$ there is $x$ with $f({x})\equiv\nu$ and $\nabla f(x)\not\equiv 0$ mod $p$, and there is a solution $\bar{x}\in \mathscr{B}$ to $g(\bar{x})=1$ with distance $\geq r$ from the real singular locus of $g(\bar{x})=1$, then $C_{n,f,\mathscr{B}}\gg_{f,r} 1$.

Take $\mathscr{B}=\frac{1}{2}\big[(\frac{1}{2s})^{1/d},(\frac{2}{s})^{1/d}\big]^k$ and $P= 2\nu^{1/d}$. If $\nu\gg_f 1$, $k>(d-1)2^d$ and $\sum P_i(x_i)\equiv\nu$ has nonsingular solutions mod $p$ then we deduce that $\sum P_i(x_i)=\nu$ has many solutions in your range.

Why does this follow from Birch's work?

That paper is about some degree $d$ forms $f_1,\dotsc,f_R$. But actually, after the introduction, we can take the $f_i$ to be general degree $d$ polynomials if we replace $f_i$ by its leading part in the following places: the definition of $f^{(i)}_{j_0,\dotsc,j_{d-1}}$ at the start of Section 2; the definition $V(\mu)$ and $V^*$ in formulae (2) and (3) of Section 3; the definition of $I(\mathscr{C};\gamma)$ in formula (9) of Section 5; and throughout Section 6.

In Section 7, Theorem 1 we need to replace $f_i$ by its leading part in the definition of $V^*$ and in the expression $\Phi[f(\bar{x})]$. The theorem above is exactly this in the case $R=1$ of a single polynomial.

The argument really does not change. A little thought to see that Lemma 5.1 is still true, that's it.

A hardcore version of the theorem above would be uniform in the lower degree parts of $f$.

Theorem? Let $f$, $g$, $P$ and $\mathscr{B}$ be as above. Suppose the polynomial $P^{-d} f(P x)$ has coefficients bounded by some fixed constant $C$. If the singular locus of $g$ has codimension $>(d-1)2^d$ in $\mathbb{C}^k$, then the number of integer solutions to $f(x)=0$ in $P\mathscr{B}$ is $$ C_{n,f,\mathscr{B}}P^{k-d}+O_{g,C}(P^{k-d-\delta}) $$ If for each $p$ there is a solution over $\mathbb{Q}_p$ with $|\nabla f(x)|_p \geq c(p)$, and there is a real solution $x\in P\mathscr{B}$ with $|\nabla f(P x)| \geq c(\infty)P^d$, then $C_{n,f,\mathscr{B}}\gg_{g,C,c} 1$. (Edit: corrected the condition at $\infty$.)

The condition on $P^{-d} f(P \bar{x})=0$ just means that if $x\in P\mathscr{D}$ then $f(x)$ is not much larger than the leading part $g(x)$, so that the lower degree parts do not dominate too badly. The alleged proof is the same as the theorem above, but you push on through sections 5 and 6 of Birch's paper with $f$ remaining inhomogeneous. I have not checked this. Modern technology might give a neater proof.

I should also mention that everything here would work for systems of several forms, with appropriate changes in the condition on the codimension of the singular locus.