One training pet to catch adder, ignoring the odds (5)
I believe the answer is:
cadet
'one training' is the definition.
'pet to catch adder ignoring the odds' is the wordplay.
'pet' becomes 'cat' (I have seen 'Common pet ' mean 'cat' so perhaps 'pet' could also mean 'cat').
'to catch' is an insertion indicator.
'ignoring the odds' means one should take alternating letters (letters in odd-numbered positions are removed).
The alternate letters of 'adder' are 'de'.
'cat' enclosing 'de' is 'CADET'.
(Other definitions for cadet that I've seen before include "Apprentice officer" , "Junior officer; younger brother" , "Younger son" , "Tomboy" , "Police force trainee" .)