Almost all common receptacles in North America are duplex - that is, there are two receptacles in one unit that mounts in a wall box. One each side (Neutral and Hot) there is a small metal bridge in the connection point between the two halves. So for normal use one connects ONE Hot wire on one side under one screw, and ONE White Neutral wire under one screw on the other side. The fact that yours has TWO White wires under adjacent screws on that side is unusual, but it MAY mean that the installer used those screws as a junction point between two Neutral wires in the box. That is NOT advised, but it gets done.
IF the intent is to have those two halves of the outlet fixture operate independently from two different Hot leads (each separately fed from breakers from different Hot bus phases), then one simply breaks off the little bridge between the halves on the HOT side only. Then the two HOT lines are connected to the separated screws on the Hot side, and a single White wire goes to the combined Neutral side that still has its bridge intact. Thus the fixture operates as two separate outlets on separate breakers, but sharing the Neutral. When this is done, the system back at the main panel uses a special DUPLEX breaker that feeds those two Hots from the two phases of the panel buses, so that if either of the circuits trips its breaker, the other ALSO trips. Thus any overload on those circuits will cut off power to BOTH of the lines into that wall box, and it is safe to work on.
With that structure, it is ALSO easy to do what OP asks about, because the two halves of the fixture CAN be separated easily. Start with: the wall box has a feed of a (Black) Hot line, a (White) Neutral line, and a bare Ground. First step is to break off that little bridge between the upper and lower halves on the HOT side. Maintain the connection of the White wire(s) to the Neutral side. Now you need a cable installed that runs out from this box to wherever the SWITCH is to be. At the switch, the two wires from that cable are attached to the two screw terminals of the switch. At the outlet box, two pigtail leads need to be connected with a wire nut to the Black Hot feed entering the box. ONE of these (MAYBE the Red lead you already have) should go to the screw for the HOT side of the outlet half you want always on - say, the lower one. The other is really the Hot (Black) wire of the NEW cable you installed going out to the switch. The WHITE wire in that cable coming back is now really the HOT return from the switch, and should be marked by taping it with black electrical tape. Then connect that lead to the HOT screw for the outlet half (say, the upper one) that is to be switch controlled. With this set up, both outlet halves are fed from a single breaker and both can be shut off entirely by that breaker in the panel. But the lower half is always on, and the upper half is controlled by the new wall switch. NOTE that there is NO change needed at the main breaker panel - no new breaker or anything.