So first of all, keep in mind the title of the Map: This is not a "US takes extra land from Mexico" Map, but is the Map of "What Polk wanted the US to expand to". In other words, it is a map of a US where Polk got everything he either tried to expand the US to, stated in some speech a desire to, or in some memoir he wanted to. In OTL, Polk offered Spain 100 Mil to just buy the Island. The implication is that in this map's timeline, Polk's offer was either big enough Spain said fuck it or Spain took the original offer and sold the island to the US.Given the sheer tiny size of the US peacetime miltary and congress sheer resistance to increasing its size there is no way they would be able to hold that much teritory. I don't think they could even if they tripled the size of the army which I sverely doubt they would.
And how exactly did they manage to get Spanish Cuba?
Because I can't imagine they could have defeated the Spanish military of the era of the war against Mexico even assuming Britain didn't decide to get involved to protect their own interests.
As for Northern Mexico? That region is at the time only somewhat more densely populated than Alta California by Mexicans...which is to say very much not all that heavily populated. The vast majority of the people living in that region were Native Americans like the Commanche(who were spread across both sides of the OTL border). I do actually see the US being able to keep control of it pretty decently.