  Let lambda_1, \lambda_2, \lambda_3, \lambda_4 be non-zero real numbers, not
all negative, with \lambda_1/\lambda_2 irrational and algebraic. Suppose that
\mathcal{V} is a well-spaced sequence and \delta >0. In this paper, it is
proved that for any \varepsilon >0, the number of v \in \mathcal{V} with v
\leqslant N for which |\lambda_1 p_1^2 + \lambda_2 p_2^3+ \lambda_3 p_3^4+
\lambda_4 p_4^5 - v| < v^{-\delta} has no solution in prime variables
p_1,p_2,p_3,p_4 does not exceed O\big(N^{\frac{359}{378} + 2\delta
+\varepsilon}\big). This result constitutes an improvement upon that of Q. W.
Mu and Z. P. Gao [12].
