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Some options:

Pick[
 lis,
 StringStartsQ[lis[[All, 1]], "a"]
 ]

{{"ab", 2, "c", 3}, {"ac", 5, "f", 6}, {"ad", 9, "c", 10}}

Select[lis, StringStartsQ[First[#], "a"] &]

{{"ab", 2, "c", 3}, {"ac", 5, "f", 6}, {"ad", 9, "c", 10}}

Cases[lis, m : {_?(StringStartsQ["a"]), ___} :> m]

{{"ab", 2, "c", 3}, {"ac", 5, "f", 6}, {"ad", 9, "c", 10}}