Beastly house by end of esplanade that's an eyesore (4)
I believe the answer is:
stye
'an eyesore' is the definition.
(I've seen this before)
'beastly house by end of esplanade' is the wordplay.
'beastly house' becomes 'sty' (I can't explain this - if you can you should give a lot more credence to this answer).
'by' means one lot of letters go next to another.
'end of' suggests the final letters.
The last letter of 'esplanade' is 'e'.
'sty'+'e'='STYE'
'that's' acts as a link.
Can you help me to learn more?
(Other definitions for stye that I've seen before include "producing eyesore" , "irritation to the viewer" , "Climb" , "Inflamed swelling on the edge of an eyelid" , "Eye sore" .)