Waring's problem with the Ramanujan $\tau$-function

We prove that for every integer $N$ the Diophantine equation $\sum_{i=1}^{74000}\tau(n_i)=N$, where $\tau(n)$ is the Ramanujan $\tau$-function, has a solution in positive integers $n_1, n_2,\dots, n_{74000}$ satisfying the condition $\max_{1\le i\le 74000}n_i\,{\ll}|N|^{2/11}+1$. We also consider similar questions in residue fields modulo a large prime $p$.