Is what the questioner did correct by completing the prayer in the North Coast—where he worked for 12 days—and shortening it in Mansoura—where his family is and he stayed for 3 days? And if it is not correct, what is its expiation?

If you have taken the North Coast as your place of residence, then completing the prayer there is valid. You are permitted to avail yourself of the concessions of travel when visiting your family in Mansoura, provided you do not intend to reside for a period that dissolves the travel status, which is four full days according to the majority of scholars. The presence of your relatives in Mansoura does not prevent you from availing yourself of the concessions; for the Companions of the Prophet, peace and blessings be upon him, used to shorten prayers in Mecca after the Hijra despite having relatives there. There is no need for you to make up past prayers. For future actions, it is more cautious to act according to the opinion of the majority of scholars.