Summary: For an integer \(p\geq 1\), let \(\Gamma_p\) be an approximative quasi-normed ideal of compact operators in a Banach space with a quasi-norm \(N_{\Gamma_p}(.)\) and the property \[ \sum\limits_{k=1}^{\infty}|\lambda_k(A)|^p\le a_p N_{\Gamma_p}^p(A)\;\;(A\in\Gamma_p), \] where \(\lambda_k(A)\) \((k=1,2,\dots)\) are the eigenvalues of \(A\) and \(a_p\) is a constant independent of \(A\). Let \(A,\tilde{A}\in\Gamma_p\) and \[ \Delta_p(A,\tilde A):= N_{\Gamma_p}(A-\tilde A) \;\exp\left[a_pb_p^p\left(1+\frac{1}{2}(N_{\Gamma_p}(A+\tilde A)+N_{\Gamma_p}(A-\tilde A))\right)^p\right], \] where \(b_p\) is the quasi-triangle constant in \(\Gamma_p\). It is proved the following result: let \(I\) be the unit operator, \(I-A^p\) be boundedly invertible and \[ \Delta_p(A,\tilde A)\exp\left[\frac{a_pN^p_{\Gamma_p}(A)}{\psi_p(A)}\right]<1, \] where \(\psi_p(A)=\operatorname{inf}_{k=1,2,\dots}|1-\lambda_k^p(A)|\). Then \(I-\tilde A^p\) is also boundedly invertible. Applications of that result to the spectrum perturbations of absolutely \(p\)-summing and absolutely \((p, 2)\) summing operators are also discussed. As examples we consider the Hille-Tamarkin integral operators and matrices.