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

Let k be a natural number and let c=2.134693\ldots be the unique real solution of the equation 2c=2+\log (5c-1) in [1,\infty) . For 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.