  We prove short-time existence for the negative $L^2$-gradient flow of the
$p$-elastic energy of curves via a minimising movement scheme. In order to
account for the degeneracy caused by the energy's invariance under curve
reparametrisations, we write the evolving curves as approximate normal graphs
over a fixed smooth curve. This enables us to establish short-time existence
and give a lower bound on the solution's lifetime that depends only on the
$W^{2,p}$-Sobolev norm of the initial data.
