= Solution
The <direct image of a coherent sheaf under a closed immersion> puts $\mathcal G=f_*\mathcal F$ in the category of coherent sheaves on $\mathbb P_k^n$. By <sheaf cohomology under a closed inclusion> and the supplied compatibility of twisting with direct image,
$$
H^p(X,\mathcal F(d))\cong H^p(\mathbb P_k^n,\mathcal G(d)).
$$
Here is a proof of the required <Serre vanishing> on projective space. A <coherent sheaf> on $\mathbb P_k^n$ is the <sheaf associated with a graded module> for a finite graded $k[t_0,\ldots,t_n]$-module. Equivalently, it has a presentation by finite sums of twisting sheaves. Use a <finite twisting resolution of a coherent sheaf on projective space>: resolve the graded module by a finite graded <free resolution>, using the <Hilbert syzygy theorem>, and sheafify; <exactness of localization> preserves the resolution. Its terms are finite sums of $\mathcal O(a)$.
Choose $d$ sufficiently large that all twists $a+d$ occurring in these finitely many terms are nonnegative. The <cohomology of twisting sheaves on projective space> then vanishes in every positive degree for every resolution term. In a short exact sequence $0\to\mathcal K_{j+1}\to\mathcal E_j\to\mathcal K_j\to0$, with $\mathcal K_0=\mathcal G$, the <long exact sequence in sheaf cohomology> identifies $H^p(\mathcal K_j(d))$ with $H^{p+1}(\mathcal K_{j+1}(d))$ for $p>0$. Iterating to the final acyclic term proves
$$
\boxed{H^p(X,\mathcal F(d))=0\quad(p>0,\ d\gg0).}
$$
The standard graded-module description used here is given in https://stacks.math.columbia.edu/tag/0BXE[Stacks Project, Section 30.15]; the vanishing follows from the displayed finite-resolution argument.
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