I am interested in the following problem: I have a finite field $F_q$, two positive integers $n>m$ and elements $a_1,...,a_m\in F_q$. How many of the polynomials $x^n+a_1x^{n-1}+...+a_mx^{n-m}+c_{m+1}x^{n-m-1}+...+c_n,c_i\in F_q$ are irreducible? What are the best known estimates, esp. for $q$ fixed and $m,n\to\infty$?

This is similar to counting irreducibles in arithmetic progressions modulo $x^m$ (once you replace $x$ by $1/x$). You can turn the problem into counting rational points on a curve (coming from a "cyclotomic function field" in the sense of Carlitz) over $F_{q^n}$ and get an estimate $q^n/n + O(gq^{n/2})$, where $g$ is the genus of the curve. Unfortunately $g$ grows like $mq^m$ so you only get good estimates for $m$ small and nothing when $m$ gets close to $n/2$. There are plenty of papers on this (e.g. by S. Cohen). There is also some experimental work by Panario et al.. Mathscinet should help you locate these.