Former deputy PM caught at last concealing housing support (5)

'former deputy pm' is the definition.
Although both the answer and definition are singular nouns, I cannot understand how one could define the other.

'caught at last concealing housing support' is the wordplay.
'caught' becomes 'cle' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'at last' indicates one should take the final letters.
'support' means one lot of letters go next to another (in a down clue, some letters hold others up).
The last letters of 'concealing housing' are 'gg'.
'cle'+'gg'='CLEGG'

Can you help me to learn more?