equation = c2[x] f''[x] + c1[x] f'[x] + c0[x] f[x] + d0[x] == 0

grid = {1.,2.,3.,4.};

c = (Sinh[1/2 (g x^4 + x^2)] + p^3 x/ (g - x^3)) g^2 x^2;

(* ... *)
p = 3.; (* actually some computation *)
g = {0.714, 0.429,0.734, 0.921}; (* some other computation *)
cComputed = Block[{x = grid}, c]
(* use this vector c to construct and solve the discretized equation *)

cPartlyComputed = g^2({1.`, 4.`, 9.`, 16.`} Sinh[{0.5`, 8.`, 40.5`, 128.`} g + {0.5`, 2.`, 4.5`, 8.`}]
+ p^3  {1.`, 8.`, 27.`, 64.`}/(g - {1.`, 8.`, 27.`, 64.`}));

cTry = c /. x -> grid;

ClearAll[CircleTimes, CirclePlus]
Attributes[CirclePlus] = Attributes[CircleTimes] = Complement[Attributes[Plus], {Listable, Protected}];
Verbatim[CircleTimes][a_] := a;
Verbatim[CirclePlus][a_] := a;
numericOrList[a_] := NumericQ[a] || Head[a] === List;
a_[CircleTimes]b_ := a*b /; numericOrList[a] && numericOrList[b];
a_[CirclePlus]b_ := a + b /; numericOrList[a] && numericOrList[b];
a_[CircleTimes](b_[CirclePlus]c_) :=
a[CircleTimes]b[CirclePlus]a[CircleTimes]c /; numericOrList[a]
a_[CircleTimes]x1_[CirclePlus]a_[CircleTimes]x2_ :=
        a[CircleTimes](x1 + x2) /; numericOrList[x1] && numericOrList[x2];


evaluate[expression_, var_, grid_] := Hold[Evaluate[
expression /. {Plus -> CirclePlus, Times -> CircleTimes} /. var -> grid
]] /. {CirclePlus -> Plus, CircleTimes -> Times}

evaluate[c,x,grid]

c2=-5832. - 432. x^2 - 1728. f x^3 - 12. (32. + 72. d1f + 144. f^2 - 27. p^4) x^4
    - 24. f (32. + 72. d1f - 27. p^4) x^5

evaluate[c2,x,grid]