There is an absolute constant D0 > 0 such that if f(x) is an integer polynomial, then there is an integer λ with |λ| ≤ D0 such that xn + f(x) + λ is irreducible over the rationals for infinitely many integers n ≥ 1. Furthermore, if deg f ≤ 25, then there is a λ with λ ∈ {−2, −1, 0, 1, 2, 3} such that xn + f(x) + λ is irreducible over the rationals for infinitely many integers n ≥ 1. These problems arise in connection with an irreducibility theorem of Andrzej Schinzel associated with coverings of integers and an irreducibility conjecture of Pál Turán. © Instytut Matematyczny PAN, 2020.