There should be one or more screws (part #12) holding the feed nut (#11) in the fixed and proper location, make sure it or those are tight. There’s also various methods of holding the cross slides feed screw (#39) in it’s fixed axial location to the cross slide itself. If it’s loose or out of adjustment, that can also greatly affect how much perceived backlash there is. There should be just barely enough clearance to allow smooth rotation of the hand wheel, but almost no axial end play. On a lot of lathes, the hand wheel is also used to help axially locate the feed screw and the amount of end play. Without the lathe in front of anyone here, it’s very difficult to be sure of what the exact cause is. It can also be a combination of more that one issue.
I firmly believe top, cross and even the carriages are meant to be taken apart, properly solvent cleaned, re-lubricated and reassembled every so often. Any moving part slowly wears in service. With a machine tool, they can also accumulate lot’s of contamination just from what there being used to do. Any time the lube oil being used is very dark or black on the slides and feed screws, that means it’s contaminated and needs to be flushed clean. It also means your not lubricating often or with enough between those solvent cleanings. Not doing so is literally wearing out the machines individual parts multiple times quicker than if there kept clean AND properly lubricated. Most at the amateur level never lubricate often or with enough. And that fresh lubricating oil serves two purposes, lubrication, and to help flush or wash out those wear particles and any contamination. George Thomas for example recommended that disassembly and cleaning after “turning cast iron” on his Myford due to the possible extra wear and damage caused by turning castings.
While it’s not impossible to make new feed nuts, most of our smaller machines use fairly small diameter feed screws and the same for the nut threads. That makes single pointing the proper imperial or metric ACME profile thread very difficult or even impossible in some cases. Without question the original manufacturer would have used the correct ACME taps for your feed nuts that few of us would have access to because of there expense. With some diameters and thread pitches, the proper ACME taps may not even be available as a standard off the shelf tap, and an extremely expensive custom ground tap would be required. If your lathe does use a common diameter and pitch, and new OEM replacement nuts are no longer a stock item, oversize feed screw nut blanks with the ACME threads already done are available from some of the better industrial level feed screw and nut manufacturers. With those, you machine the exterior to the correct size and duplicate the original mounting method. It can still be a touchy job to use those though. The center line of the nut threads have to properly align with the center line of the feed screws exact location, so they require some fairly accurate measurements and machining.
For some machine tools, the feed screws bearing or bushing hanger just behind the hand wheel aren’t doweled for there vertical or side to side location. With those, you leave the mounting screws slightly loose and withdraw the side all the way back towards the operator until it stops. Then you final tighten those mounting screws. That helps self align that bearing or bushing to the feed screws nut C/L. Not doing it in that way will almost always create binding and very high rates of wear on the feed nuts and feed screw threads. Mechanical issues always have a cause and a correct solution if your willing to take some time to properly understand how the assembly is supposed to work when it’s correct, and analyze what the faults could or might be using some logic when it isn’t.