Short answer: the fingering can be the problem, but there are other possibilities, one of which is finger placement.
Congratulation for your study of the soprano recorder. Hope you are having fun.
-> The fingering you show in your question is a typical fingering for a baroque soprano recorder.
I presume you have a baroque/english-model recorder because a school/"modern" recorder has often only one hole instead of two smaller ones for each of the ring and auricular fingers (last two bottom holes of the picture), and you would have noticed that. But first check this before going on.
The "modern" or "german" model fingering would use only the index for the right hand (left hand is the same) for a F on a soprano. You can spot a "german" model by its quite small fifth hole (corresponding to the semitone between F and E).
There are a few things to check if the fingering problem is out of the way:
Hope it helps. A lot of people dismiss the recorder as an "easy" instrument for kids. But there is no "easy" instrument.