Although robust learning and local differential privacy are both widely studied fields of research, combining the two settings is just starting to be explored. We consider the problem of estimating a discrete distribution in total variation from $n$ contaminated data batches under a local differential privacy constraint. A fraction $1-ε$ of the batches contain $k$ i.i.d. samples drawn from a discrete distribution $p$ over $d$ elements. To protect the users' privacy, each of the samples is privatized using an $α$-locally differentially private mechanism. The remaining $εn $ batches are an adversarial contamination. The minimax rate of estimation under contamination alone, with no privacy, is known to be $ε/\sqrt{k}+\sqrt{d/kn}$, up to a $\sqrt{\log(1/ε)}$ factor. Under the privacy constraint alone, the minimax rate of estimation is $\sqrt{d^2/α^2 kn}$. We show that combining the two constraints leads to a minimax estimation rate of $ε\sqrt{d/α^2 k}+\sqrt{d^2/α^2 kn}$ up to a $\sqrt{\log(1/ε)}$ factor, larger than the sum of the two separate rates. We provide a polynomial-time algorithm achieving this bound, as well as a matching information theoretic lower bound.