My situation: I hurt my wrist and was at the hospital all night and I missed Isha -1 (prayer) When I got home, I was in so much pain and made tayammum (dry ablution once then I prayed Isha and Fajr. I later realised that I was able to make Wudu as normal (with water) but did not realise that at the time. And I only made dry ablution once for two obligatory prayers. Do I have to make up for those two prayers? Please give some reference from Qur'an and Sunnah.

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

Dear brother,

Allah Says (what means): {But if you are ill or on a journey or one of you comes from the place of relieving himself or you have contacted women and do not find water, then seek clean earth and wipe over your faces and hands with it. Allah does not intend to make difficulty for you, but He intends to purify you and complete His favor upon you that you may be grateful.} [Quran 5:6]

The scholars interpreted the illness mentioned in the verse and which makes lawful to perform Tayammum (dry ablution) instead of ablution as the fear of getting sick, or the fear that the illness gets worse, or the fear of delaying its cure.
So, if you thought when you made Tayammum that ablution could worsen your case, or delay your cure, or have some bad side effects, then you are not required to make up the prayers you had performed with Tayammum.

Otherwise you should now make your ablution and make up the two prayers that you had performed with Tayammum.
For more benefit, please refer to Fatwa 31285.

Allah knows best.