I know Nazi Germany was planning on eventually annexing Fascist Italy, but they would need time to recuperate and plan after the war.

Perhaps you would like to share the source of that scholarship? The claim is fantastical.

As for the question, surrounding joint axis war aims. There were no wide reaching ones agreed. For example Italy took actions such as its decision to join the War against France and Britain as well as the invasion of Albania & Greece unilaterally.

The armistices that ended the Battle of France resulted in very little permeant territorial loss - only Alsace-Lorraine. Italy gained access to French port and rail facilities in Djibouti, maintaining the occupation it currently conducted of Melton & Corsica. But these territories were not transferred de jure to Italy. And there was no question of Italy gaining control of Savoy, Nice or even Tunisia - long states Italian ambitions. So italy gained very little because Germany opposed it.

Which brings me on nearly to Axis war aims,

The main take home from the stuff above is that

• there were no clear war aims or plans events took off independently of each other
• both countries had separate war aims that I will detail now

Italy’s long term ambition was to break British and French strategic hegemony over the Mediterranean.

That would mean control over or a equal share in the governance and operation of the Suez Canal. Control over colonial possessions such as Malta, Tunisia, Djibouti as well as areas which contained Italian populations that were important due to nationalism such as Nice/Savoy and importantly the Dalmatian coast of Yugoslavia (which Italy had fought and won WW1 for only to be denied by Versailles resulting in what Italians call the “vittoria mutilata” mutilated victory).

There is however a lot more debate amongst historians surrounding German war aims.

Roughly there are three separate perspectives; globalists who claim that the post war aims were to impose a new world order benefiting Germany a world order would be one that would the challenge United States, this is separate to continentalists who claim that the war aims were limited to creating a European order & still others such as AP.Taylor who argue that there was no clear or coherent plan and that it is simply opportunism. There is evidence to support and contradict each perspective.