I failed to take my RMD in 2021. I will aske for a waiver for the penalty, but I wondered if I can claim the RMD which I requested and received on my 2021 return?

No.  The RMD must be reported in the year you actually received the distribution.  If that is 2022, you must report on your 2022 tax return.  The date of distribution required was December 31st.

I read that I could take the RMD up until April 2022, and I received my distribution on February 25, 2022. I haven't received a 1099R for this distribution for 2021.  Is that why I have to wait and claim it for 2022 along with a second RMD for 2022?