Irreducible polynomials with prescribed sums of coefficients

Let $q$ be a power of a prime, let ${\mathbb{F}}_q$ be the finite field with $q$ elements and let $n\ge 2$. For a polynomial $h(x)\in {\mathbb{F}}_q[x]$ of degree $n\in \mathbb{N}$ and a subset $W \subseteq [0,n] := {0, 1, \ldots, n}$,wedefinethesum-of-digitsfunction$$S_W(h) = \sum_{w \in W}[x^{w}] h(x)$$tobethesumofallthecoefficientsof$x^w$in$h(x)$with$w \in W$. In the case when $q=2$, we prove, except for a few genuine exceptions, that for any $c\in {\mathbb{F}}_2$ and any $W\subseteq [0,n]$ there exists an irreducible polynomial $P(x)$ of degree $n$ over ${\mathbb{F}}_2$ such that $S_W(P)=c$. In particular, restricting ourselves to the case when $# W = 1$, we obtain a new proof of the Hansen-Mullen irreducibility conjecture (now a theorem) in the case when $q=2$. In the case of $q>2$, we prove that, for any $c\in {\mathbb{F}}_q$, any $n\ge 2$ and any $W\subseteq [0,n]$, there exists an irreducible polynomial $P(x)$ of degree $n$ such that $S_{W}(P) \neq c$.