  In this paper, we first study the perturbations and expressions for the
generalized inverses $a^{(2)}_{p,q}$, $a^{(1, 2)}_{p,q}$, $a^{(2, l)}_{p,q}$
and $a^{(l)}_{p,q}$ with prescribed idempotents $p$ and $q$. Then, we
investigate the general perturbation analysis and error estimate for some of
these generalized inverses when $p,\,q$ and $a$ also have some small
perturbations.
