If the map is north to south, you can't have a temperate area in the south. Your continent is roughly 5000 km in height. It's almost half of North America. If the north is cold, it could be around the polar circle (66th) and the south close to the tropic (21th).
You could have a forest in the south but it's a bit too cold to be a jungle. More like a wet and dry forest like southern China maybe?
The ''what?'' area is probably dry for the most part and could be move closer to the tropics.
The area in the peninsula is not much different from the surroundings. In mid latitudes, dominant winds go from west to east and they would be able to carry the moisture over all the landmass since it's 1000 km across or something like that. But the eastern part is probably somewhat drier. But that is debatable since we also have to consider the effect of the Jet stream.
The peninsula could have a ''lake effect'' like the Great lakes meaning an increase in precipitation eastward. It's a very erratic effect and limited is area coverage though, it's not very significant. The peninsula also temperate the climate but the effect if very small.
Plain is more likely a forest unless it was chopped down.
The desert in the north, it would need to be almost on the pole in order to be considered as a desert (ice desert).
So the progression would look like this:
The climate in the south would be like Burkina Faso with a small strip of jungle. Then it's a savanna that turns into a desert. Eventually, it becomes colder and more humid and there is more vegetation with some steppes and Mediterranean climates in the west not far from the ''hot''. The temperate area would be covered by a temperate forest like in France. Going north, it's getting colder but the coast could stay relatively warm like Norway. The northern tip turns into a tundra.