Pattern matching multiple symbols names and powers

I believe that if you want to match a Symbol by its constituent characters that you do need to use string patterns, therefore your solution is reasonable. I recommend a variation:

p2 = s_Symbol^_. /; StringMatchQ[SymbolName[s], "V*"]

s_Symbol makes sure that the expression that is passed to SymbolName is actually a Symbol. I also find it somewhat more readable. Test:

MatchQ[p2] /@ {V1, V1^2, V2, x^2, y}   (* v10 operator form syntax *)

{True, True, True, False, False}

The use of Condition rather than PatternTest makes it easier if you may be working with held expressions and need to add Unevaluated:

p3 = s_Symbol^_. /; StringMatchQ[SymbolName[Unevaluated@s], "V*"]

This allows:

V1 = "Fail!";
Cases[Hold[V1, V1^2, V2, x^2, y], x : p2 :> Hold[x]]
V1 =.;

{Hold[V1], Hold[V1^2], Hold[V2]}

To make the code a bit prettier consider an abstraction such as:

symPat = s_Symbol /; StringMatchQ[SymbolName[Unevaluated@s], #] &;

Then simply:

Cases[{V1, V1^2, V2, x^2, y}, symPat["V*"]^_.]

{V1, V1^2, V2}