I suspect the plan was basically to give Pope a very big army and then conduct the overland campaign that Lincoln always wanted. Whether Pope knew this is actually an interesting question because as Daoust points out Pope had every means to ask for 6th Corps earlier than he actually had need of it, and did not;
Apparently aware. On the 21st of August Gen. Pope was indeed informed by Halleck's HQ that he would be reinforced as the Army of the Potomac units came up...
he also didn't retire behind the Bull Run to combine the armies between the 29th and 30th August, despite being aware it was an option. He may have wanted to avoid sharing the glory he thought was to come.
As early the 22nd of August Gen. Halleck was encouraging Pope to attack where he saw an opportunity...
Pope was again instructed on the 26th to attack once he found Lee's army...
On the same date, Franklin's sixth corps got its marching orders to join Pope... but for various reasons it did not join Pope's army until after the battle...
Here's a basic rundown of the time line on General Franklin's Sixth Corps, which the Administration and General-in-Chief Halleck desired to have joined with General Pope's Army before the battle...
Anyways, here's a sample of Pope's orders from Halleck by the close of the 27th... keep line with Washington open, troops from the AoP etc. expected to come out to by it, etc.
Something I should note in this connection is that (while I think Welles might not necessarily have known this, and possibly Lincoln didn't remember it) McClellan had actually sought and obtained permission from Lincoln to write a letter about things that were not strictly military but which related to the conduct of the war; this became the Harrisons Landing letter.
My understanding is that Lincoln was not particularly offended about Mac giving him the Harrison's landing letter; just thanked him and put it in his pocket, etc. The President's secretaries Hay and Nicolay thought he was more alarmed at McClellan's particular views about his military situation.
General McClellan reported the same...
Hay and Nicolay say Lincoln was actually offended some at Mac's lack of enthusiasm relative to pressing reinforcements to Pope before battle at Second Manassas was joined...
There's also the weirdness with respect to advancing from Harrisons Landing which is that McClellan's latest instructions from Lincoln were actually to wait for reinforcement, those reinforcements being Burnside's force (which never arrived). In fact, and relevantly to the point about Burnside which you made later, it is my understanding that everyone in McClellan's high command structure was under the impression (because Burnside had said so) that he was due to arrive on the 21st July.
The reason Burnside hadn't arrived prior to that point is that he was told not to join McClellan until directed to (from Washington).
Instead Burnside was summoned to Washington and offered AoP command, which he refused.
Mac commanded the Army of the Potomac. He had stepped down from the position of General-in-chief to command it in person on the Peninsula. He therefore had no authority to give orders to Burnside in North Carolina, or anyone outside of the Army of the Potomac particularly. Lincoln noted in a speech in August the McClellan had every right to call for such reinforcements as he thought necessary, and the Secretary of War to provide him just as many as he, the President saw fit.
J.R. Hay says President Lincoln was aware of "staff talk" in the Army which could commence with rumors and end with disobedience where it created confusion among the officers. Burnside had his orders. McClellan had his orders. Pope had his orders. But there is only one commander-in-chief, and he decides what the orders are, but he relies upon the generals to carry them out somehow.
Case in point, before the Seven Day's battle about Richmond erupted Gen. Burnside was evidently planning a movement inland, I think to Goldsboro, but got orders to proceed to the James River at the close of June. Several thousand troops were embarked, but about July 2 General Burnside was apparently informed, by "staff talk" rumors, through Col. Hawkins, that McClellan had taken Richmond after heavy fighting. So he presumed it necessary to delay his departure, presuming he would shortly receive notice to debark and continue operations toward Goldsboro. Veterans recalled...
By the 5th however, after McClellan was whipped and re-established at Harrison's landing, Gen. Burnside informed the War Dept. that he had received information of McClellan's defeat and withdrawal to the James River. A large part of his forces were then carried to Newport News, VA, within supporting distance of McClellan's army, and formed into the Ninth Army Corps. It moved to Aquia Creek in the first days of August. Even if it had joined Mac at Harrison's it was only ca. 10,000 strong, and Mac wanted at the least double that number to renew an offensive.
General Halleck assumed the general-in-chiefdom on July 11 to get the Army under some control, and to find out if McClellan would make an advance from Harrison's landing, otherwise to remove the troops...
So the above fixes August 3 as the date that the Administration and Army was
certain General McClellan was never going to make a forward movement from Harrison's landing, as he was then
ordered to abandon it and carry the army by water to Aquia Creek to promote operations between Washington and Richmond.
August 4, Gen. McClellan once notified that he was to withdraw the army responded with alarm...
Halleck responded that it would not be rescinded, so he aught to get moving. And asked why he hadn't already sent off his sick, which had previously been ordered some days before the withdrawal order...
Did the fate of the Union hang of McClellan retaining his toe-hold perimeter on the Peninsula at Harrison's landing after August 4, 1862? The Government apparently didn't think so. It thought it was more important to put those troops in the field of battle developing in front of Washington.
Like I say, I think it's possible here that Lincoln forgot he'd explicitly given McClellan permission for the Harrisons Landing letter, or that he forgot that he'd told McClellan to wait for reinforcement before advancing (specifically McClellan's preference was to move on Petersburg) but it does look like attempts to replace McClellan with Someone Else are a key feature; if you consider the formation of the Army of Virginia to be another of these then it goes back prior to the Seven Days.
Lincoln probably only cared about the Harrison's landing letter, to the extent it offended his cabinet, many of whom felt McClellan had to go months before it was written.
In the post-Antietam period dealing with General McClellan was again contentious in the common manner. Hay notes in his diary the President stated he had come to suspect, fairly or not, that Mac just really didn't want to "hurt" the rebs enough.