Two first-half goals – the opener after just 25 seconds – left the Bully Wee with a mountain to climb against their Championship opponents in their second Premier Sports Cup group match.
And although Ally Love pulled one back after the break, the Ayrshire side left Broadwood with all three points.
Lennon said his side paid the price for carelessness in possession right from the kick-off.
He said: “It was poor. You get possession and three times within the first 30 seconds we've given the ball away.
"They are ruthless, they punish you and it really settles them down. It was tough enough before.
"They go again and they're huffing and puffing a little bit and I thought we were careless once again with a lot of possession, asked nowhere near enough questions in the first half, didn't get anywhere near in terms of the way we set up.
"We spoke before about getting personal and tight with them and once again it was our own doing [in the lead-up to the second goal], overdoing things in our own half.
"We've got to recognise that at times you can't go in there and play suicidal football, particularly when you're not passing the ball well in the first place.
"Sometimes you've got to bend a little bit there.
"We regrouped at half-time and I thought we were a lot better.
"They didn't get it all one-way traffic, I thought there were wee spells in the game where they dropped off and we asked questions of them.
"There's no doubt you would expect a Championship club and a club of Kilmarnock's history just coming down to have opportunities and I thought David Mitchell had one or two fantastic saves to keep us in it.
"At that stage of the game we want to have a go and with the personnel we had we tried to be as positive as we could.
"There were a lot of lessons to be learned today in terms of the first half."
Clyde were scheduled to be in action again last night (Tuesday), away to Lowland League side East Kilbride, before travelling to Morton for their final group match on Saturday.