  Majorana zero modes have been shown to be the simplest quasiparticles
exhibiting pure odd-frequency pairing, an effect that has so far been
theoretically established in the static regime. In this work we investigate the
formation of Majorana modes and odd-frequency pairing in $p$-wave
spin-polarized superconductors under a time-dependent drive. We first show that
the driven system hosts multiple Majorana modes emerging at zero and $\pi$,
whose formation can be controlled by an appropriate tuning of the drive
frequency and chemical potential. Then we explore the induced pair correlations
and find that odd-frequency spin-polarized $s$-wave pairing is broadly induced,
acquiring large values in the presence of Majorana modes. We discover that,
while odd-frequency pairing is proportional to $\sim1/\omega$ in the presence
of Majorana zero modes, it is proportional to $\sim 1/(\omega-\pi\hbar/T)$ in
the presence of Majorana $\pi$ modes, where $T$ is the periodicity of the
drive. Furthermore, we find that the amount of odd-frequency pairing becomes
larger when multiple Majorana modes appear but the overall divergent profile as
a function of frequency remains. Our work thus paves the way for understanding
the emergent pair correlations in driven topological superconductors
