Is there any appropriate way to offer cash to a lecturer or TA for private tutoring?
At my university in the USA, it would likely be illegal for the instructor or the TAs of the particular class to participate in this. There is just too much of a (perceived) conflict of interest since they are evaluating you for a grade for the class.

However, there would not be anything barring you from approaching a graduate student who was a former TA for the class; nor would there be a problem in approaching an instructor at another institution or someone who is at your institution who is not teaching your section of your class that term. You could even ask an upperclass student who took the class last year to tutor you.

Your department secretary may be able to connect you with a grad student who needs a few extra dollars and isn't in the conflict-of-interest position to tutor you directly. Or post something in your local craiglist or student bulletin board system.