And here's the simple answer, that you probably expect your students to find...

In[21]:= Solve[eqn, c]//TrigToExp//FullSimplify//InputForm                      

Out[21]//InputForm= {{c -> -2/3 + 8/(9*Pi)}}

Solve considers the expressions declared as variables to be atomic objects. Effectively they are first replaced with new symbols and then solved for:

In[1]:= f[x, y] ==                                                              
    Sqrt[1 - x^2 - y^2] -                                                       
      8/Pi*Integrate[                                                           
         f[x, y], {x, 0, 1/2}, {y, -Sqrt[1/4 - x^2] + 1/2,                      
           Sqrt[1/4 - x^2] + 1/2}] /. f[x, y]->z // InputForm                   

Out[1]//InputForm= z == Sqrt[1 - x^2 - y^2] - z

In[2]:= Solve[%, z]//InputForm                                                  

Out[2]//InputForm= {{z -> Sqrt[1 - x^2 - y^2]/2}}

This is similar to

In[3]:= Solve[x==Sin[x], Sin[x]]

Out[3]= {{Sin[x] -> x}}

If you specify {x, y} as the variables then f[x, y] is treated as a function of the variables and Solve knows that it cannot solve the equation.

In[4]:= Solve[f[x, y] ==
   Sqrt[1 - x^2 - y^2] -
    8/Pi*Integrate[
      f[x, y], {x, 0, 1/2}, {y, -Sqrt[1/4 - x^2] + 1/2,
       Sqrt[1/4 - x^2] + 1/2}], {x, y}]//Head

Solve::nsmet: This system cannot be solved with the methods available to Solve.
    Try Reduce or FindInstance instead.

Out[4]= Solve

The answer of Solve is rubbish, and the reason may be that you fed it a integral functional equation, for which Solve is not prepared.

Here is a possible derivation of the solution:

reg = ImplicitRegion[x^2 + y^2 <= y && x >= 0, {x, y}];
eq = f[x, y] == Sqrt[1 - x^2 - y^2] -
   8/Pi*Inactive[Integrate][f[x, y], Element[{x, y}, reg]]
generalFormOfF = f[x, y] :> Sqrt[1 - x^2 - y^2] + c
eq /. generalFormOfF
% // Simplify
% /. Inactive[Integrate][a_ + b_, w__] :>
  Inactive[Integrate][a, w] + Inactive[Integrate][b, w]
% /. Inactive[Integrate][c, w__] :> Integrate[c, w]
Solve[%, c]
f[x, y] == Sqrt[1 - x^2 - y^2] + c /. %[[1]]
% /.
 Inactive@Integrate -> NIntegrate

Unfortunately, the exact value of the integral seems to take a very long time.