Too much information is almost always better than not enough
I'm assuming that you're saying it's failing the MOT inspection due to excessive play in the ball joints. Maybe you could explain to us over in the US what the MOT inspection is and what happens when your car fails inspection.
How many miles are on the car, and what kind of driving is it used for? Is it mostly high speed motorway and in-town on good roads, or small country roads that haven't been resurfaced since before the war?
If the ball joints are truly worn beyond the acceptable limit, about all you can do is to replace them.
Which ball joints in particular are we talking about?
Where and how are they measuring the excess play?
How much play does the MOT inspector say that it has, and how much do they allow?
Is there any recourse that allows you to take it to another inspection station if you disagree with the first one?
Have you talked to your local dealer's service department and asked them whether it was likely the ball joints were worn with the use and mileage on the car? If so, what was their response?
With a bit more information I'm sure someone on this forum can help you get the right answer.