NFHS rule 4-9-4 says that the clock should be started on the throw-in when the ball touches or is touched by a player. It doesn't say "legally" touched, just touched.
By rule, the error for the missed 1/1 can be corrected until the end of the first dead ball after the clock started. The ball became live on the throw-in for the kicked ball violation as soon as it was handed to or placed at the disposal of the thrower- rule 6-1-2(b). Iow, yup, it's too late by rule to correct the error at that time.
Note rule 5-9-1 also says that it doesn't matter either whether the official signalled "clock start" on the kicked-ball throw-in or not. The timer is authorized to start the clock using 4-9-4 without having any signal from the official.
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