Interpolating irregularly missing data points of regularly spaced data
I found a method similar to the zero-padding FFT interpolation but in which the missing samples can be nonuniformly spaced. I think it exactly solves the problem you have. The method is described here:

J. Selva, "FFT interpolation from nonuniform samples lying in a regular grid", IEEE Trans. on Signal Processing, vol. 63, n. 11, June 2015.

The code is available here.