pts = {{-0.2, -0.2}, {0.2, 0.2}};
NIntegrate[Nearest[pts-> "Distance", {x, y}] , {x, y} [Element] Disk[]]
(*Nearest: The default distance does not give a real numeric distance when
applied to the point pair {x,y} and {-0.2,-0.2}.*)

pts = {{-0.2, -0.2}, {0.2, 0.2}};
Block[{f, nf},
 nf = Nearest[pts -> "Distance"]; (* precompute NearestFunction *)
 f[x_?NumericQ, y_?NumericQ] :=   (* NumericQ prevents evaluation on symbolic x, y *)
   First@nf[{x, y}];              (* Nearest returns a list; extract distance with First *)
 NIntegrate[f[x, y], {x, y} ∈ Disk[]]
 ]
(*  1.69082  *)