Young pet: one's anxious to have many (6)
I believe the answer is:
kitten
'young' is the definition.
(kitten is a kind of young)
'anxious to have many' is the wordplay.
'anxious' becomes 'kit' (I can't explain this - if you can you should believe this answer much more).
'to have' means one lot of letters go next to another.
'many' becomes 'ten' (example).
'kit'+'ten'='KITTEN'
'pet one's' acts as a link.
I am not very happy about this link. Some or all of it may belong to another bit of the clue.
Can you help me to learn more?
(Other definitions for kitten that I've seen before include "Young rabbit" , "Young of a beaver" , "Mini-mog?" , "Immature cat" , "Baby rabbit" .)