  We investigate the accelerator stage of a Plasma-Modulated Plasma Accelerator
(P-MoPA) [Phys. Rev. Lett. 127, 184801 (2021)] using both the paraxial wave
equation and particle-in-cell (PIC) simulations. We show that adjusting the
laser and plasma parameters of the modulator stage of a P-MoPA allows the
temporal profile of pulses within the pulse train to be controlled, which in
turn allows the wake amplitude in the accelerator stage to be as much as 72%
larger than that generated by a plasma beat-wave accelerator with the same
total drive laser energy. Our analysis shows that Rosenbluth-Liu detuning is
unimportant in a P-MoPA if the number of pulses in the train is less than
$\sim$ 30, and that this detuning is also partially counteracted by increased
red-shifting, and hence increased pulse spacing, towards the back of the train.
An analysis of transverse mode oscillations of the driving pulse train is found
to be in good agreement with 2D PIC simulations. PIC simulations demonstrating
energy gains of $\sim$ 1.5 GeV ($\sim$ 2.5 GeV) for drive pulse energies of 2.4
J (5.0 J) are presented. Our results suggest that P-MoPAs driven by few-joule,
picosecond pulses, such as those provided by high-repetition-rate thin-disk
lasers, could accelerate electron bunches to multi-GeV energies at pulse
repetition rates in the kilohertz range.
