What is the ruling on one who swore an oath upon the Mushaf but did not tell the whole truth, having changed the intention of the oath inwardly to avoid embarrassment and severing ties of kinship? And what is the expiation for him?

Revealing a man's flaws is not forbidden backbiting; rather, it is an obligatory counsel. It is not permissible for a Muslim to lie or swear falsely except out of necessity or for a legitimate Sharia interest, such as averting hardship or severance of kinship ties, provided that dissimulation (tawriya) is used.