Let , . Assume that f does not have repeated roots. Assume as well that, for every prime q, mod q2 has at least one solution in . Then, under these two necessary conditions, there are infinitely many primes p such that f(p) is square-free.
"Square-free values of f(p), f cubic." Acta Math. 213 (1) 107 - 135, 2014. https://doi.org/10.1007/s11511-014-0117-2