For a prime $p$ and $a_1,\dotsc,a_n\in\mathbb F_p^\times$, consider the system of equations $$ \begin{cases} \begin{align} a_1 + \dotsb + a_n &= 0 \\ a_1x_1 + \dotsb + a_nx_n &= 0 \\ \qquad &\ \vdots \\ a_1x_1^K + \dotsb + a_nx_n^K &= 0 \end{align} \end{cases} $$ How large can $K$ be given that this system has a solution in the variables $x_1,\dotsc,x_n\in\mathbb F_p^\times$?

By the Chevalley–Warning theorem, if $n>K(K+1)/2$, then there are solutions with at least some of the variables distinct from $0$. Is it true, say, that if $K>C\sqrt n$ with a suitable absolute constant $C$, then the system does not have non-zero solutions? (For my purposes, it would be fine to know that there are no non-zero solutions with at least $2n/3$ among $x_1,\dotsc,x_n$ pairwise distinct.)

An essentially equivalent restatement of the problem is as follows:

What is the smallest possible number of non-zero coefficients of a polynomial $P\in\mathbb F_p[x]$ of degree $\deg P\le p-1$ given that $P$ has a root of multiplicity $K$ at $x=1$?

There are many *similar* results and problems around (the Tarry-Escott problem, Linnik lemma, Vinogradov system, Borwein-Erdelyi-Kos results for polynomials over the integers, etc), and I suspect that the problem stated above has also been studied. Any references will be appreciated.