On the quantity $m^2-p^k$ where $p^k{m}^2$ is an odd perfect number -- Part II

Let $p^k{m}^2$ be an odd perfect number with special prime $p$. Extending previous work of the authors, we prove that the inequality $m<p^k$ follows from $m^2-p^k=2^{r}t$, where $r\ge 2$ and $\gcd (2,t)=1$, under the following hypotheses: (a) $m>t>2^r$, or (b) $m>2^r>t$. We also prove that the estimate $m^2-p^k>2m$ holds. We can also improve this unconditional estimate to $m^2-p^k>313m^2\mathrm{/}315$.