Partitio Numerorum: sums of a prime and a number of $k$-th powers

Let $k$ be a natural number and let $c=2.134693\dots$ be the unique real solution of the equation $2c=2+\log (5c-1)$ in $[1,\mathrm{\infty })$. Then, when $s\ge ck+4$, we establish an asymptotic lower bound of the expected order of magnitude for the number of representations of a large positive integer as the sum of one prime and $s$ positive integral $k$-th powers.