  Recently, Schlage-Puchta proved super multiplicity of $p$-deficiency for
normal subgroups of $p$-power index. We extend this result to all normal
subgroups of finite index. We then use the methods of the proof to show that
some groups with non-positive $p$-deficiency have virtually positive
$p$-deficiency. We also compute the $p$-deficiency in some cases such as
Fuchsian groups and study related invariants: the lower and upper absolute
$p$-homology gradients and the $p$-Euler characteristic.
