Hopes this is working without Anthony's last idea (10)
I believe the answer is:
hypothesis
'idea' is the definition.
(hypothesis is a kind of idea)
'hopes this is working without anthony's last' is the wordplay.
'is working' indicates anagramming the letters.
'without' means one lot of letters goes inside another ('without' can be similar in meaning to 'outside').
'last' indicates one should take the final letters.
The final letter of 'anthony' is 'y'.
'hopes'+'this'='hopesthis'
'hopesthis' is an anagram of 'hpothesis'.
'hpothesis' placed around 'y' is 'HYPOTHESIS'.
(Other definitions for hypothesis that I've seen before include "A tentative theory" , "conjecture" , "What's supposed" , "Supposition" , "idea" .)