  In this paper we introduce the notion of a quasi-powerful $p$-group for odd
primes $p$. These are the finite $p$-groups $G$ such that $G/Z(G)$ is powerful
in the sense of Lubotzky and Mann. We show that this large family of groups
shares many of the same properties as powerful $p$-groups. For example, we show
that they have a regular power structure, and we generalise a result of
Fern\'andez-Alcober on the order of commutators in powerful $p$-groups to this
larger family of groups. We also obtain a bound on the number of generators of
a subgroup of a quasi-powerful $p$-group, expressed in terms of the number of
generators of the group. We give an infinite family of examples which
demonstrates this bound is close to best possible.
