On the sum of a prime the square of a prime and the k-th power of a prime

In this paper we prove that, when k is odd, with at most O(N(1/2-)2/5x4(k-1)x k + epsilon) exceptions, all positive odd integers n <= N can be written as sums of a prime and a square of prime and a k-th power of prime; when k is even, with at most O (N(1/2-)2/5x4(k-1)x k + epsilon) exceptions, all positive odd integers n <= N with n equivalent to 0 or 1 (mod3), can be written as sums of a prime and a square of prime and a k-th power of prime.