Before removing the spherical portion, potential at point $$P$$(Center of cavity)

$${V_{sphere}}\,\, = {{ - GM} \over {2{R^3}}}\left[ {3{R^2} - {{\left( {{R \over 2}} \right)}^2}} \right]$$

$$ = {{ - GM} \over {2{R^3}}}\left( {{{11{R^2}} \over 4}} \right) = - 11{{GM} \over {8R}}$$

Mass of removed part = $${M \over {{4 \over 3}\pi {R^3}}} \times {4 \over 3}\pi {\left( {{R \over 2}} \right)^2}$$ = $${M \over 8}$$

Due to cavity part potential at point $$P$$

$${V_{cavity}}\,\,$$ = $$ - {{G{M \over 8}} \over {2{{\left( {{R \over 2}} \right)}^3}}}\left[ {3{{\left( {{R \over 2}} \right)}^2} - {0^2}} \right]$$

           = $$ - {{3GM} \over {8R}}$$

So potential at P due to remaining part,

$$ = {V_{sphere}}\,\, - \,\,{V_{cavity}}\,\,$$

$$ = \,\, - {{11GM} \over {8R}} - \left( { - {3 \over 8}{{GM} \over R}} \right)$$

$$ = {{ - GM} \over R}$$

Note : Potential inside the sphere of radius R and at a distance r from the center,

V = $$ - {{GM} \over {2{R^3}}}\left[ {3{R^2} - {r^2}} \right]$$

Here M = mass of the sphere, r = distance from the center of the sphere