Core problem: Attackers run out of Momentum and spawn too far back. Defenders keep a cheap spawn plus mortars and field gun next to the Headquarters and farm you while you walk. The Support Objective is the usual answer to both problems. It is optional, but it is the thing that makes HQ takeable.

1What the Support Objective Actually Gives

Forward spawn for whoever holds it, usually a better jump-off than the default line.
Buildable support weapons once you own it: Toffee Apple trench mortar and a field gun aimed at the Headquarters fight.
Momentum for attackers on first capture. One of the ways to get Momentum back when you are low.
Cannot be locked. Either team can recapture it for the rest of the sector.
HQ is the only thing that advances the map. Support is the thing that makes HQ takeable. It is not required to unlock the next sector.

2Decision: Take It First or Go Straight at HQ?

Which of the two points is HQ vs Support is random each sector. Look at the map before you commit the first wave.

Take Support first (default)
When Momentum is mid/low or HQ is a meatgrinder
Cap it, spawn on it, build the mortar and gun, then hit HQ from a flank with fire support. Several players say they struggle to crack HQ without that fire support. After you cap Support, immediately push parallel to it and wipe defender beacons and caches so they cannot spawn behind you.
As defender
Recap Support even if you are not using the guns
Denying the attacker spawn and mortar is the point. Contest it the moment they leave it thinly held.

3Concrete Attacker Sequence

4Common Mistakes

If Momentum is already gone and HQ is untouched, Support is usually the only thing that still lets you stay in the fight. If HQ is one build away from locked, finish HQ first.

5Videos: Support Then HQ Loop and Attacker Beacons