Diophantine approximation with four squares and one kth power of primes

Let k be an integer with $$k\ge 3$$ and $$\eta $$ be any real number. Suppose that $$\lambda _1, \lambda _2, \lambda _3, \lambda _4, \mu $$ are non-zero real numbers, not all of the same sign and $$\lambda _1/\lambda _2$$ is irrational. It is proved that the inequality $$|\lambda _1p_1^2+\lambda _2p_2^2+\lambda _3p_3^2+\lambda _4p_4^2+\mu p_5^k+\eta |<(\max \ p_j)^{-\sigma }$$ has infinitely many solutions in prime variables $$p_1, p_2, \ldots , p_5$$ , where $$0<\sigma <\frac{1}{16}$$ for $$k=3,\ 0<\sigma <\frac{5}{3k2^k}$$ for $$4\le k\le 5$$ and $$0<\sigma <\frac{40}{21k2^k}$$ for $$k\ge 6$$ . This gives an improvement of an earlier result.