Abstract It is a folklore conjecture that the Möbius function exhibits cancellation on shifted primes; that is, $\sum_{p{\,\leqslant} X}\mu(p+h) \ = \ o(\pi(X))$ as $X\to\infty$ for any fixed shift h > 0. This appears in print at least since Hildebrand in 1989. We prove the conjecture on average for shifts $h{\,\leqslant} H$, provided $\log H/\log\log X\to\infty$. We also obtain results for shifts of prime k-tuples, and for higher correlations of Möbius with von Mangoldt and divisor functions. Our argument combines sieve methods with a refinement of Matomäki, Radziwiłł and Tao’s work on an averaged form of Chowla’s conjecture.