It doesn't appear that option is available.
chess.com clock app?
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Yeah, I've never heard of that being an option.
A simple way for you to accomplish that is for you to resign if you see you've spent more than the minute.
I would suggest making these games unrated (or unrated takeback), however. Doing that on a rated game could look like you're trying to manipulate ratings.
@omnipaul, I believe the question is about the actual clock app on mobile, where that isn't an issue since you use it OTB. Probably should be an option, if not just for the reason of adding time for illegal moves 
Ah, I wasn't aware there was such an app.
Still, my suggestion for how to accomplish this holds. It just means they can ignore my third paragraph.
As to adding time for illegal moves, perhaps they could use some larger time (such as 10 minutes), and then just mentally change the point at which they "run out of time."
I downloaded the app recently to play some quick games before the site I was at closed and I didn't have my clock with me. Not too bad if no one bangs on the phone and good for more casual games where you still want to enforce a time limit.
I watched a TV programme, where Carlson played one of the presenters at blitz and he gave himself 30 seconds to the presenters 2 minutes. I just think this is a good idea to give weaker players a chance against stronger opponents. I've tried doing it on an iPad with Shredder and Hiarcs, but you can't edit the clocks on these apps either. Regular chess clocks do have this function though.
I guess due to the imbalance and lost fairness caused by a different start time could be a reason for such a feature not existing for now. But your explanation makes sense in order to rebalance a thought advantage ... maybe the support forum would be a good place to ask for such a feature?
Our chess club used to do it when pitting the A team players against the B C and D team players. 10 minutes for the A team against 15 minutes. I only request these handicap controls for fun, not as a serious option in tournaments.
How to add time in the app while the opponent play illeagal move
If you want a free and easy-to-use chess clock with all the most common and used features, you can download this android application on your phone: https://play.google.com/store/apps/details?id=giulio.di.maria.chessclock.google.
Regular updates will be released and suggestions or personal features are accepted and welcomed!
Hope you'll enjoy it! 
I'd like to play my son by giving myself a time handicap on the chess.com chess clocks. I'd like to give him 5 minutes and me 1 minute. How do I do this? I've tried custom, but can only seem to find various equal set times for both players? anyone?