Create rule whose LHS is already defined

I suggest wrapping the LHS a[t] in the Rule you are trying to return in HoldPattern:

Clear[func]
func[k_] := Block[{a},
   a[t_] := t^k;
   {HoldPattern[a[t]] -> a[t]}
 ]

func[2]
(* Out: {HoldPattern[a[t]] -> t^2} *)

a[t] /. func[5]
(* Out:  t^5 *)

func1[k_] := ReleaseHold @ Block[{a}, a[t_] := t^k; {Hold[a[t]] -> a[t]}]
func2[k_] := Activate @ Block[{a}, a[t_] := t^k; {Inactive[a[t]] -> a[t]}]

func1[3]

{a[t] -> t^3}

func2[3]

{a[t] -> t^3}

a[t] /. func1[5]

t^5

a[t] /. func2[x] 

t^x

Could return pure functions instead:

pow[k_] := a -> Function[t, t^k]

Examples:

a[t] /. pow[5]
a[10] /. pow[x]
a[2] /. pow[3]

t^5

10^x

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