At home, tire badly; beginning of afternoon's lethargy (7)
I believe the answer is:
inertia
'lethargy' is the definition.
(thesaurus)
'at home tire badly beginning of afternoon's' is the wordplay.
'at home' becomes 'in' ('in' can be similar in meaning to 'at home').
'badly' indicates an anagram.
'beginning of' suggests taking the first letters.
The initial letter of 'afternoons' is 'a'.
'tire' anagrammed gives 'erti'.
'in'+'erti'+'a'='INERTIA'
(Other definitions for inertia that I've seen before include "inclination not to budge" , "Resistance to motion" , "Apathetic state" , "Tendency to remain at rest or in steady motion" , "Passiveness" .)