It is permissible to pray with clean shoes on. The Prophet ﷺ prayed in his shoes on multiple occasions, as authentically reported in both Ṣaḥīḥ al-Bukhārī and Ṣaḥīḥ Muslim. Companions such as Anas ibn Mālik (may Allah be pleased with him) narrated that the Prophet ﷺ regularly prayed wearing his sandals.
The condition is that the shoes must be free of any impurity (najāsah). If there is any doubt about cleanliness, the shoes should be removed out of precaution and respect for the prayer.
In conclusion, wearing shoes during prayer is not only permissible but an established Sunnah when the conditions of cleanliness are met.