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}}