Not sure exactly of the current strat/supply system, but as far as HQ goes it must be fully destroyed to have any beneficial impact on the game. Full destruction will cause complete loss of radar for the enemy. Partial destruction will reduce dot/bar radar to just bar dar.
The HQ is resupplied by the City via road/train convoys. If the City is destroyed before HQ then it will remain down for a longer period of time. HQ can be resupplied tho, so unless the nearby bases are capped it is only an interim measure.
Well, that's how it used to work anyway...
The AH 1.11 readme is as follows:-
Adjusted factory and city down times.
Cities are now down for 3 hours.
Factories are 3 hours.
Player supplies now provide 15 minutes of down time.
Convoys & Trains supply 30 mins.