It has been shown that long-range p-wave superconductivity in a Kitaev chain can be engineered via an ac field with a high frequency [Benito et al., Phys. Rev. B 90, 205127 (2014)]. For its experimental realization, however, theoretical understanding of Floquet engineering with a broader range of driving frequencies becomes important. In this work, focusing on the ac-driven tunneling interactions of a Kitaev chain, we investigate effects from the leading correction to the high-frequency limit

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... the emergent p-wave superconductivity. Importantly, we find new engineered long-range p-wave pairing interactions that can significantly alter the ones in the high-frequency limit at long interaction ranges. We also find that the leading correction additionally generates nearest-neighbor p-wave pairing interactions with a renormalized pairing energy, long-range tunneling interactions, and in particular multiple pairs of Floquet Majorana edge states that are destroyed in the high- frequency limit.