Should one repeat the prayer if a small part of washing the foot was left out during ablution due to haste, and this was remembered upon entering the mosque and the prayer being established?

You must repeat your ablution (Wudu') and prayer, because this purification is not valid due to its incompleteness, and the prayer is not valid due to thement of the condition of purity. In the Hadith, the Prophet, peace and blessings be upon him, "saw a man praying, and on the top of his foot was a spot the size of a dirham that water had not touched. The Prophet, peace and blessings be upon him, commanded him to repeat his ablution and prayer."