Minimal polynomials of elements of order $p$ in $p$-modular projective representations of alternating groups

Let $F$ be an algebraically closed field of characteristic $p>0$ and let $G$ be a quasi-simple group with $G/Z(G)\cong A_n$. We describe the minimal polynomials of elements of order $p$ in irreducible representations of $G$ over $F$. If $p=2$, we determine the minimal polynomials of elements of order $4$ in $2$-modular irreducible representations of $A_{n}$, $S_n$, $3\cdot A_6$, $3\cdot S_6$, $3\cdot A_7$, and $3\cdot S_7$.