Diophantine approximation with one prime, two squares of primes and one kth power of a prime

Let $1<14/5$, $lambda_1,lambda_2,lambda_3$ and $lambda_4$ be non-zero real numbers, not all of the same sign such that $lambda_1/lambda_2$ is irrational and let $omega$ be a real number. We prove that the inequality $|lambda_1p_1+lambda_2p_2^2+lambda_3p_3^2+lambda_4p_4^k-omega|le (max (p_1,p_2^2,p_3^2,p_4^k))^{-psi(k)+arepsilon}$ has infinitely many solutions in prime variables $p_1,p_2,p_3,p_4$ for any $arepsilon>0$ where $psi(k)=minleft(rac1{14},rac{14-5k}{28k} ight)$.