I want to give examples to explain want I want to know first. Let $G colon s mapsto G(s)$ be a PRG.

Let $F_{1} colon s mapsto G(s) Vert b$, where $b = bigoplus_{k = 1}^{G(s)} G(s)(k)$. Obviously, $F_{1}$ is not a PRG, since the last bit can be predicted. But if we remove the last bit, it is a pseudorandom string.

Let $F_{2} colon s mapsto G(s) Vert G(s)$, $F_{2}$ cannot be PRG, too. but if only choose $G(s)$ bits properly, it can be pseudorandom.
So, if $F colon {0,1}^l to {0,1}^n$ is a polytime deterministic generator. Perhaps, $F$ is not a PRG, but $F$ may be closed to a PRG. Namely, there exists a compression algorithm $A colon {0,1}^{n} to {0,1}^m$, where $n geq m > l$, such that $A circ F$ is a PRG. Is it welldefined? Can we use something like $n – min_{A} m$ to measure how is $F$ closed to PRGs? Is there a standard way to measure the maximum random bits of all the outputs of $F$?