Letk=GF(q) be the finite field of orderq. Letf1(x),f2(x)∈k[x] be monic relatively prime polynomials satisfyingn=degf1>degf2⩾0 andf1(x)/f2(x)≠g1(xp)/g2(xp) for anyg1(x),g2(x)∈k[x]. WriteQ(x)=f1(x)+tf2(x) and letKbe the splitting field ofQ(x) overk(t). LetGbe the Galois group ofKoverk(t).Gcan be regarded as a subgroup ofSn. For any cycle patternλofSn, letπλ(f1,f2,q) be the number of square-free polynomials of the formf1(x)−αf2(x) (α∈k) with factor patternλ(corresponding in the natural way to cycle pattern). We give general and precise bounds forπλ(f1,f2,q), thus providing an explicit version of the estimates for the distribution of polynomials with prescribed factorisation established by S. D. Cohen in 1970. For an application of this result, we show that, ifq⩾4, there is a (finite or infinite) sequencea0,a1,…∈k, whose length exceeds 0.5logq/loglogq, such that for eachn⩾1, the polynomialfn(x)=a0+a1x+…+anxn∈k[x] is an irreducible polynomial of degreen. This resolves in one direction a problem of Mullen and Shparlinski that is an analogue of an unanswered number-theoretical question of A. van der Poorten.