What is the concise, Sharia-compliant way to return a wife after an irrevocable divorce, and is it permissible to pray for facilitating her marriage and divorce in order to have her back?

One should not rush into divorce to avoid harm. If a man divorces his wife with the third pronouncement, she is not lawful for him until she marries another husband, as Allah the Almighty says: (And if he has divorced her [for the third time], then she is not lawful to him afterward until she marries another husband) [Al-Baqarah/230]. Tahlil marriage (marriage for the purpose of making a divorced woman lawful for her previous husband) is forbidden and invalid. The Prophet (peace be upon him) cursed the one who performs tahlil and the one for whom tahlil is performed. There is no other way to return a thrice-divorced wife except for her to marry another man and then be divorced by him. It is recommended to supplicate to Allah to facilitate marriage to her if it is good, and to entrust the matter to Allah regarding the separation of her current husband if her return is good. It is advised to sever attachment to her and seek another wife, unless it is decreed for her to marry and be divorced by her other husband, and both spouses believe they will observe the limits of Allah.