Pop wants gradually to remove Thatcher at the beginning, that prickly character (3,5-6)

'prickly character' is the definition.
I don't know anything about this answer so I can't judge whether it can be defined by this definition.

'pop wants gradually to remove thatcher at the beginning' is the wordplay.
'pop' becomes 'Iggy' (musician Iggy Pop).
'wants' means one lot of letters go next to another.
'gradually to remove' becomes 'winkle' (winkle can mean to extract or remove).
'thatcher' becomes 'Mrs T' (Margaret Thatcher was nicknamed Mrs T).
'at the beginning' says to put letters next to each other (some letters go at the beginning of others).
'iggy'+'winkle' is 'iggywinkle'.
'iggywinkle' after 'mrst' is 'MRS TIGGY-WINKLE'.

'that' is the link.

Can you help me to learn more?