Will adding a superconducting ring inside solenoid increases the inductance of it? Consider a solenoid having $N$ turns and cylindrical length, a superconducting ring is placed inside having its plane perpendicular to axis of solenoid, if ring (having self-inductance $L$) is very small we can consider $r<<R$ of solenoid and one can assume magnetic field to be going inside the ring to be uniform, but will the inductance of solenoid get changed decrease/increaae by how much. My question is what happens such that inductance change happens how to show it?
 A: A superconducting loop as you described it, will deflect all the magnetic flux directed on its $πr^2$ loop area, around the loop towards the inner walls of the larger radius R solenoid. The squeezed flux in a smaller volume inside the solenoid area will therefore increase the strength $B$ of the field around the superconductive loop inside the solenoid around the location where the superconductive loop is.
Through the sc loop area there is no flux going through therefore there is no magnetic field $B$ on the surface area of the sc loop. It will disturb the field in that area and make it no homogeneous. However the flux $Φ$ through the total inner volume of the solenoid will remain unchanged for a given current I of the solenoid for the reasons explained previously.
The inductance of the solenoid is given by:
$$L=\frac{\mu N^{2} A}{\ell}$$
where $l$ the length of the solenoid and $A$ its cross-sectional area and μ the magnetic permeability of the medium inside the solenoid. In your case the sc loop will increase the permeability in that area around the sc loop concentrating the flux towards the inner wall of the solenoid and acting like ferromagnetic core inserted. Thus, increasing locally the inductance of the coil in that location. Effectively, it would be like adding more turns $N$ overall to the solenoid.
The excluded area $dA$ from the sc loop is not a problem since no magnetic flux is lost but just diverted around the sc loop $B=\mu n l$.
Expect, with a sc loop inserted at the center of the solenoid to increase the overall inductance $L$ of the solenoid since it acts like a ferromagnetic core or ferrite inserted.
Notes: I assumed that your question assumes that the superconductive loop has already a persistent current $I_{sc}$ flowing inside prior the solenoid is powered up by an external emf source? Therefore the Meissner effect:

If this is not the case and you are referring to first an inactive sc loop material inserted in an inactive solenoid and then powering up the solenoid, then the situation changes and magnetic flux will pass through the sc loop normally and will be trapped there even after the solenoid is powered down.

In this second case, concerning the overall inductance of the solenoid for $r<<R$ nothing will really change.
source: http://web.pdx.edu/~pmoeck/lectures/312/supercon.pdf
