Let Rk(n) denote the number of representations of a natural number n as the sum of three cubes and a kth power. In this paper, we show that R3(n) [Lt] n5/9+ε, and that R4(n) [Lt] n47/90+ε, where ε > 0 is arbitrary. This extends work of Hooley concerning sums of four cubes, to the case of sums of mixed powers. To achieve these bounds, we use a variant of the Selberg sieve method introduced by Hooley to study sums of two kth powers, and we also use various exponential sum estimates.