Bloomer by an egghead, reading and writing nought but arithmetic! (5)
I believe the answer is:
error
'bloomer' is the definition.
(I've seen this before)
'an egghead reading and writing nought but arithmetic' is the wordplay.
'an egghead' becomes 'e' (head letter of 'egg').
'reading' becomes 'r' (one of the traditional three R's of education).
'and' means one lot of letters go next to another.
'writing nought but' becomes 'ro' (I can't explain this - if you can you should believe this answer much more).
'arithmetic' becomes 'r' (one of the traditional three R's of education).
'e'+'r'+'ro'+'r'='ERROR'
'by' is the link.
Can you help me to learn more?
(Other definitions for error that I've seen before include "Mistake or mishap" , "Solecism" , "Howler" , "Trial and ___" , "A mistake or inaccuracy" .)