Firstly check that it's the swivel and not the wheel bearing, lots of garagges really don't know what they are talking about!
Assuming yours does then the swivel is supported by 2 taper roller bearings if it's a non-abs axle and a lower TRB and upper Railko bush on ABS axles. The former can be adjusted by removing shims that control the pre-load (cruch) on the tapers, but if it's now loose one or both might be collapsing so think about replacing them. If there is any oiol left in the swivle has it started to leak?
If it's the ABS axle then you can still adjust it but you cannot do anything about sidewyas play in the upper bush other than replace the bush and pin, not cheap or easy.