Is there any word for "small hand written note/letter" in English?
There is a commonly used phrase borrowed from French.
A Billet Doux is a hand written love note.
Frequently pronounced Billy Doo when used this way but often pronounced in the French way.
- a message
- a note, a short note
message (noun) A usually short communication transmitted by words, signals, or other means from one person to another. TFD
note (noun) "a brief informal letter" TFD
You could say "I'm sending her a note of appreciation".