Is it permissible to travel to Medina with the intention of visiting the Prophet's grave in addition to the intention of visiting the Prophet's Mosque and praying therein?

It is a Sunnah for whoever intends to travel to Medina to intend to visit the Prophet's Mosque and pray therein, due to what was narrated from the Prophet, peace and blessings be upon him: "A prayer in this mosque of mine is better than a thousand prayers in any other mosque, except Al-Masjid Al-Haram." After that, it is desirable to visit the grave of the Prophet, peace and blessings be upon him. However, undertaking a journey solely for the purpose of visiting the grave is not permissible. If the intention of the journey is to visit the Mosque and pray therein, then visiting the grave as a secondary purpose is permissible. But if the primary intention is to visit the grave, and praying in the Mosque is secondary, then this is not permissible.