How does single-slit interference affect a double-slit experiment?

Answer 1

What you call "single slit interfernce" is just the way the light spreads out from a single slit due to diffraction.

In a double-slit experiment, there is interfence between the two separate beams diffracting from each slit. Light spreads out at an angle determined by single-slit diffraction, but there are light and dark bands due to two-slit interference

Answer 2

Each double slit is in itself a single slit. That's why you see a "modulating" single slit pattern over the double slit pattern