"Cut in" is a phrasal verb meaning to interrupt someone who is speaking.
Sentence Meaning:
Ray cut in while his mother was talking to her friends.
→ Ray interrupted his mother while she was talking to her friends.
Option Analysis:
A. cut in ✅ = interrupted
B. went after ❌ = followed/chased
C. looked forward ❌ = awaited eagerly (usually followed by to)
D. put out ❌ = extinguished or caused inconvenience