What will happen in Young's double slit experiment, when instead of screen I put a black screen with a hole and a second screen behind the black one. Will it still form interference pattern albeit a faint one or no interference pattern will be observed?
If the hole in the black screen is small enough, you will not see any interference pattern, but (faint) single slit diffraction at the end. If the hole you refer to is significantly larger, it will simply mask part of the original interference pattern.
You can think of this in terms of Huygens–Fresnel principle, where the every point of light coming out of the hole becomes a source of a spherical wave. The location of this hole along the interference pattern only determines the intensity of light (and thus of the subsequent pattern).
From your description, I think you are talking about the diffraction pattern from a pinhole, and that is an Airy Disk. https://en.wikipedia.org/wiki/Airy_disk It shows a central maximum with surrounding rings which decrease in intensity as you get further from the center of the pattern.