How can I effectively play Covenant?

I think I may understand the problem that you are having. You say you are overpowered by the number of enemy units, but that is normal whilst using the Covenant.

Switching from playing as the UNSC to the Covenant is difficult. The difference between having a leader/ using leader powers requires different strategies.

You may have noticed that humans have a lower max population, because they have stronger units. Their advantage is their strength, hitting hard and fast and decimating the other armies. Using basic units (having Scorpions vs Wraiths for example), the humans will usually come out victorious with a full army vs a full army, despite having fewer units.

In order to use the covenant effectively, you need to make use of the transport pads that automatically move units at your base to the leader. Covenant units will die quickly, and you need reinforcements quickly. Having a higher max unit count allows that losing the first ones will not make that much of a difference, but as soon as they start dropping you need to replenish your units by creating more at one of your bases.

It helps to put the base rally flag straight onto the transport pad, because as soon as units are created they will transport to the leader and straight into the battle (as opposed to human units which need to be flown in on pelicans or transported there manually).

In order to effectively use this strategy, there is one thing that is required.

Don't let your leader die!

It takes a very long time for them to rebuild, and the battle will usually be over before they are. And it halts any progress you may have made against enemies even if you do beat them. Units can't transport whilst the leader is dead.

You can have your leader in the battle, but make sure they stay alive. Make sure to have at least a couple of Engineer units (from the Summit building) for the Brute Chieftain or the Arbiter (If you've not yet got the upgrade that replenishes his health every time he kills a unit in Rage mode. Even if you do it's still a good idea to get 1 or 2).

Having the Prophet with the Shield of the Forerunner upgrade is very useful, because he can back out of a fight, and be back in very quickly once his shield is replenished. It is also much easier to get units, particularly slow ones like infantry units, to where you want to attack once he has his flight chair, as you can fly him right up to an enemy base and start dropping in units mustered at your own base to destroy it before the enemy army can react.

It's best to have units transport in from more than one base, as they will need to queue to use the transporter if there are too many trying to transport from a single pad.

Also watch out for the UNSC leader power that stops all leader powers in a location. This will stop units coming through the transporter as well.

I'll briefly address your other points as well.

In order to use this tactic, you need to have a high rate of production. Build plenty of warehouses. Try to claim at least 2 bases, and have at least 3/4 upgraded warehouses on each one, and at least one shield generator, filling the rest with unit production buildings (or Temple at the main base). Once you have this level of production, it should be reasonably easy to replenish your army as it gets depleted in battle.

As the Covenant are weaker, it is better to focus more on building counter units (Jackals, Hunters and Vampires) to fight whatever the enemy is throwing at you. Using Banshees vs Hornets/Hawks or Wraiths vs Scorpions, the Covenant will almost always be destroyed first.

Hunters are probably the best Covenant unit to use against UNSC, as two of their counter units are vehicles, and they are pretty good against Marines/ODSTs. However, even fully upgraded Hunters will struggle against Scorpion/Grizzly units, as they are much faster, but Banshees are very good against Scorpions. Back up the Hunters with some Jackals if they have Flamethrowers, as the Jackals have a much better range. Hunters can't shoot flying units, so having a few Vampires helps.

When fighting other Covenant, Jackals are very good (especially with the Beam Rifle upgrade), as they can be used to quickly take out the enemy leader, making sure they can't get reinforcements. Again, Jackals aren't great against air units, so some Vampires always help. Banshees will usually beat all of the vehicles that the Covenant have, so don't bother with Hunters against them as they will slow down your army advancements.

Finally, always get a handful of Engineers, not just for the leader but also to go to work on your other units. They also only use a single population spot, and have a high health, so they work as a good distraction to draw away enemy fire from your main units, and they will float there and heal each other making them even harder to kill.

Hope these tips help!

If they don't, just build a Scarab and 30 Engineers to heal it. If you can hold off your enemy for as long as it takes to get the resources and build it, this is one of the more fun tactics, as it will be pretty much unstoppable to everything except a full army of Hunters/Cobras/Hawks/Grizzlys.