They can both be correct, depending on how you phrase the sentence, and what you are talking about.
There are many cases where a patient lacks the required knowledge yet they/she work/works it out.
There are many cases where patients lack the required knowledge yet they work it out.
In the first sentence, we are referring to 1 individual at a time, with many examples of a single individual. You can use they/she, and work/works. However, depending on whether or not every individual you are talking about is a she, it may be best to stick in the 3rd person plural. They/work.
In the second sentence, we are referring to multiple patients, or a case including a group of patients (at one time). In this case again, 3rd person plural is the best option.
Also, commas, and completing the thought.
There are many cases where patients lack the required knowledge , yet they work it out in the end.