Operation Trappenjagd ended on May 19th historically, and thus the freeing up of VIII Air Corps would probably allow for a quicker liquidation of 57th Army in the end stages of Second Kharkov. This frees up 1. Gebirgs-Division and 100. leichte Infanterie-Division sooner, as well as reducing the burden on the Romanians and the Italians, enabling for a quicker and more effective encirclement of the Barvenkovo salient. German/Axis losses will obviously be less, while the escapees from the salient will be far fewer; Soviet sources claim 22,000 men escaped the pocket-almost assuredly an exaggeration-but here it is doubtful if more than a handful will. Operating under the assumption Manstein's 11th AOK is kept in the South sans its siege train while VIII Air Corps remains too, it is likely Kleist in Operation Fridericus II will succeed in his goal of not only bagging the Soviet 9th Army but also Moskalenko’s 38th Army as was intended.
To the North, with the siege train of 11th AOK, Army Group North will probably be forced into conducting Operation Bettelstab against the Oranienbaum bridgehead. Once that is liquidated, Operation Moorbrand against the against the Pogost’e salient can occur, during which or soon after the four or five 300 series Divisions will have arrived, enabling their use against either the Volkhov (If the situation by that point is still conducive for such) or for use directly against Leningrad in Operation Nordlicht. Regaradless of which direction of attack is used, the successful completion of such results in the loss of supply to Leningrad, ensuring its inevitable surrender no later than early 1943 but most likely sometime in the fall as starvation sets in.
This creates immediate problems for STAVKA and Stalin, especially once Fall Blau sets off and it will be more successful here given the better position of AGS from our Alt-ending in Second Kharkov. Stalin historically was convinced the German main effort would be against Moscow, not changing his opinion until after Voronezh and even then he kept the Soviet main effort in 1942-the disastrous Operation Mars-focused against Army Group Center. Thus, the two strong, strategic offensives by AGN and AGS would serve to essentially overload Soviet defensive capabilities, as they could only afford to effectively respond to one or the other without weakening their forces opposing AGC. Losing Leningrad or the vital strategic industries of the South would be a blow the USSR could ill afford in 1942, with the latter in particular offering decisive end results for the Germans vis-a-vis the Soviets. Stalin could, of course, abandon his focus against Army Group Center in order to achieve the forces necessary to respond to both, but that results in a wash because it frees the Germans to do the same by drawing forces away from AGC to respond to the situations on the other fronts. Alternatively, AGC still has the jumping off points for operations against Moscow and if the Soviets have sufficiently diluted the Fronts opposing them, there is nothing to stop AGC from seizing the initiative and attempting a new drive on the Soviet capitol.
Outside of the immediate battles of the Eastern Front, wider geo-political issues would be underway. The larger disaster of Second Kharkov and its aftermath, followed shortly by Fall Blau, the surrender of Sevastopol and then the effective encirclement of Leningrad would probably crystalize opinion in Anglo-American circles that the USSR is in collapse.
Marshall was already calling for a Japan First strategy anyway since London had vetoed an invasion of France and as the OTL situation developed, JPS 43 was created,
‘‘Strategic Policy of the United Nations and the United States on the Collapse of Russia,’’ Aug. 7–8, 1942. The assessment was that, in the event of a Russian collapse,
they'd abandon all offensive actions against Germany; this would include TORCH. According to
American Popular Opinion and the War Against Germany: The Issue of Negotiated Peace, 1942 by Richard W. Steele,
The Journal of American History , Dec., 1978, Vol. 65, No. 3 (Dec., 1978), pp. 704-723, American popular opinion was also firmly in support of such:
On concentrating on the Japanese, the question asked was: "Granting that it is important for us to fight the Axis every place we can, which do you think is more important for the United States to do right now: put most of our effort into fighting Japan or put most of our effort into fighting Germany?" 62 percent responded Japan, and 21 percent Germany. Bureau of Intelligence, OFF, "Survey of Intelligence Materials," No. 21 (April 29, 1942), PSF "OWI," Roosevelt Papers.