I would like to contact my supervisor after a long time and I do not know what to say to him, knowing that I have not finished the part that I am supposed to have finished. indeed, because of personal problems (my father had an accident, I had health problems) I could not advance at the programmed rate. I think I should contact my supervisor and tell him that I'm sorry for the delay and that I'm going to send him the work in 2 weeks but I do not know how to say that in a formal and professional way. Help please
"Dear Professor, I am sorry to have to inform you that a number of personal problems have disrupted my work programme. I am now fully engaged again on my work [if that is true] and hope to send you [say what it will be] shortly. [ I advise you against giving a specific date: hostage to fortune]. I should be happy to explain in more detail and answer any questions that you may have."
The basic idea is to state clearly that there was a problem and what you are doing about it but without making too many promises.
I agree with Solar Mike, at this point formality might not be the most important concern here.
I just had a big break, and recently returned to work with my supervisors, my first email to my supervisors after months of radio silence was simply
"Dear professor *, hope you are well. I have recovered and am ready to resume my research, I'd like to schedule a supervision meeting soon to catch up and plan the next steps. Regards."
For your supervisor, the message is "the student is resuming" and that's it, save the details (reasons why you were away from your study) for a face to face meeting.
Remember your supervisor is always there to support you. Don't over think it.