I just had this exact same situation. The client restricted me to 2 hours per week. They gave me a task that resulted in 2 hours by Friday with additional work to go. I told them that I wold finish this task because there was some learning time invovled but next week, I needed to keep my effort to the 2 hours and couldn't do as many tasks or we owuld need to increase the number of hours avalible. If you are clear and have great communication, it should help resolve the issue positively on both sides.
Thanks Jennifer, I'll log the time manually and talk to them next week - it's a good project and I'm grateful that they offered it to me, hopefully they'll appreciate the extra effort I'm making to complete it for them
To answer your initial question, can you log the additional hours and will the client get charged for that? No, they will not get charge as the contract is set to 10hrs max. Any additional will just be logged under overtime.
Here is the longer version of your question. One, I understand your dilemma, but this is an example we all see often from newer freelancers about client always requesting for more or not wanting to pay for additional time. It always starts with letting fly the first time and not communicating with them out of fear to lose a job or bad feedback, whatever the fear is. Secondly, It's ok to say to your client that the job will take longer than the x amount of hours placed on the contract (usually better to clarify this before accepting the contract). Then if you want to further clarify that you will do it anyways then note "I will take responsibility in this and finish per our agreement (essentially my loss not theirs)". instead of just letting it be out of fear you handled it like a professional through responsibility, communication, and honesty. Also handy if ask for additional revisions and your past 10hrs paid + unpaid x hrs overtime it took to finish. Also, take out the 20% fee.
In another situation, let's say you end up having a long-term relationship with this client and want the same task done which they come back with the same previous amount. The client thinks perfect I found this person who can it do it for this specific amount. On the other hand, you know from experience it will take much longer. This is another example where not communicating the first time becomes an issue.
In the end, you should do what you think its best for a successful project. I would just tell you from a client perspective, honesty and communication is far better than thinking I got a 'good deal'