The Beatles actually opened for Bruce Channel just before they hit it big.. and as a harmonica player myself, I can tell you John Lennon was right to be embarrassed
Of all the Lennon on harmonica songs, I think "Love Me Do" was the one most influenced by "Hey! Baby." Interestingly, by the time John entered his "Dylan" period in 1964-1965, he more or less put down the instrument, which of course was one of Dylan's trademarks.
Off the top of my head, here are some of the songs Lennon played harmonica on...
Love Me Do
Little Child
Please Please Me
Chains
Misery
There's a Place
From Me To You
Thank You Girl
One After 909 (first version, circa 1963--appears on
Anthology Vol. 1)
I'll Get You
I'm a Loser
I Should Have Known Better
Rocky Raccoon
These are officially released songs, not counting various outtakes, live cuts, etc.