assalamu alaikum wa rahmatullahi wa barakatuh “I went to the masjid to pray Asr and they were already praying, so I joined the jama‘ah. After I made up the rak‘āt I missed, can I still pray the 2‑rak‘ah masjid sunnah? And if the same thing happens for Fajr, what should I do?”

Ustadh AbdulAziz al-Haqqan

You kean tahiyat al masjid, no u don't pray it.