Abstract. We introduce a new inductive approach to the lace expansion, and apply it to prove Gaussian behaviour for the weakly self-avoiding walk on ${\Bbb Z}^d$ where loops of length $m$ are penalised by a factor $e^{-\beta/m^{p}}$ ($0<\beta \ll 1$) when:\\ (1) $d>4$, $p \geq 0$; \\ (2) $d \leq 4$, $p > \frac{4-d}{2}$. \\ In particular, we derive results first obtained by Brydges and Spencer (and revisited by other authors) for the case $d>4$, $p=0$. In addition, we prove a local central limit theorem, with the exception of the case $d>4$, $p=0$.