  We improve the large sieve inequality with $k$th-power moduli, for all $k\ge
4$. Our method relates these inequalities to a restricted variant of Waring's
problem. Firstly, we input a classical divisor bound on the number of
representations of a positive integer as a sum of two $k$th-powers. Secondly,
we input a recent and general result of Wooley on mean values of exponential
sums. Lastly, we state a conditional result, based on the conjectural
Hardy-Littlewood formula for the number of representations of a large positive
integer as a sum of $k+1$ $k$th-powers.
