Three hundred Space Marines cannot kill a million Orks. But they can kill a thousand Orks a thousand times.
That is the principle with which the Adeptus Astartes wage war. Concentration of force inflicting defeat in detail. You do not have to fight every Ork at once, only the ones in front of you.
And it is also the strategy you employ against Snikrot's Orks, Demi-Company taskforces plunging into the equatorial jungles on search and destroy missions. A massed aerial bombardment hits the Snakebite fortress: the Orks will rebuild the damage in short order, but at least it will keep them occupied.
The Steel Dragons advance under the cover of Land Speeders and gunships, blasting everything they come across before moving on. Ork patrols are slaughtered, outposts demolished and spore-infested undergrowth put to the torch. You do not stop to hold ground, overwhelming the greenskin response with raw speed. By the time they muster in numbers against you, you are already a hundred kilometers away, attacking another target. And when they move to pursue, another task force hits them from behind.
The outlying areas are easy prey, but as the spearheads penetrate deeper into the jungle the density of Orks grows ever greater, and so too does intensity of the fighting. Where the Steel Dragons risk being bogged down you assign gunships on fire support missions, Thunderhawks dropping incendiary payloads and spraying Ork hordes with underwing heavy bolters. And where your men risk being overwhelmed, you extract them by air and take advantage of the now out of position Orks to hit a different, vulnerable area. Even so, you have yet to breach into the heart of the forest, and that is where the true fighting will be had.
You take to the field personally several times, commanding a number of different task forces, but in truth your most valuable contribution is your handling of the campaign's logistics. A force of Astartes this large on a purgation mission consume ammunition more voraciously than entire Regiments of Imperial Guard, and resupply runs must be organized precisely to keep up the tempo. For this, you must coordinate the delivery timetables of fresh ammunition from Voss-Prime, as well as Thunderhawk or Drop Pod deliveries into the deep jungle.
It is during this time that you examine the possibilities of obtaining additional support for your campaign. The Steel Dragons can kill as many Orks as you like, but you cannot hope to exert control over the jungle long-term. Ideally, you would cull the greenskin numbers enough that the ADF could later step in to handle suppressing and eventually eradicating the xenos over the course of many decades or even centuries, but it always pays off to start the process as early as possible.
The other idea that crosses your mind is that with negligible air assets or defences, the greenskins would be a perfect target for a few Navy Bomber or Gunship Wings. You've been exploiting the same weakness yourselves, but you only have so many Thunderhawks, and they must handle supply runs and transportation in addition to providing air support.
[Rolled 75, 71, 28]
Fortunately, it seems that Raukov has taken your assignment of Pra'Tar and Second Company to the Ash Wastes as a matter of personal honour. He cannot allow himself to be seen as being outdone by the Steel Dragons, and so he has assigned several Regiments of the ADF to your aid. It is in truth a token force, but even a few depleted and exhausted Regiments go a long way towards securing your back lines and ensuring the Orks do not creep back into areas already put to the flame.
[Rolled 68, 15, 27]
The Navy is… more difficult. The Administratum still smarts from being outvoted on the matter of Armageddon as well as losing one of the Munitorum's major supply worlds, and this attitude filters down the ranks. Lord-Admiral Ishmael Kuzmathoff is Parol's replacement as the commander of Battlefleet Armageddon, and by all accounts the man knows exactly who signs his paychecks. You know without having to ask that to gain his support, even for a relatively small and low-risk engagement, will cost you dearly.
Still, there is another option. Despite Parol's sidelining, you still have a broad base of support among the Admirals and Vice-Admirals of the Battlefleet, and they have the authority to order a deployment without going through Kuzmathoff. You would have to time it well, when the Lord Admiral is off-system, and it would likely worsen relations with the man further when he returns to find out about it, but it might just be worth it to end the Ork threat on Armageddon once and for all.
[] Formally petition Battlefleet Armageddon for aid against the Orks. You will owe Lord-Admiral Kuzmathoff 1-2 Major Favors.
[] Go behind Kuzmathoff's back to have Air Wings assigned to the jungle campaign. Will worsen relationship with Kuzmathoff from Suspect to Bad.
[] You cannot afford to be indebted to or in bad terms with the Navy, and will make do without them.