I'll give you the best advice in a situation like that, 4 weeks away from the MOT!
Don't go spending a lot of money on it at this stage. The rot you mention is non-structural & will only fail the test if it has rusty bits of metal on view & sticking out with ragged, sharp edges. The thing to do is to bodge it to make it tidy for the time being. Stick the piece of trim back on to cover up the rust - ie, double sided tape, two or three 'clods' of body filler, nuts & bolts or self tapping screws - (try not to make too much of a mess of the trim as they are hard to come by now & VERY expensive new - £70 ish & you may want to fit it properly later).
The bits you should be looking at & worrying about are the structural bits underneath!!!
If it's gone on the curved bit of chassis that comes down from the top of the rear spring mount (best seen when you take the rear wheel off), then you will have to dismantle half of the rear suspension (if it's gone really bad), to get at it to weld it properly!!
In most cases, when a Carlton goes in the above place badly, it's £40 for the MOT - which you then get back from the local scrap yard when you have to take it for it's last 'one-way' journey

Rugbyfans car has just failed on this bit & unless you really love your car or can do the work yourself, it's the end of the road

Also, check the front chassis rail, especially on the drivers side - on top, under it, behind the bumper on the side under the wing.......& the worst bit - down the side of the steering box near the box mounting where the inner wing joins the top of the chassis rail - again, it's a one way trip to the scrap yard if it's gone really bad here, too

If you're 4 weeks from the test, then you must be within the month for taking it in & getting 13 months test if it passes..........So don't mess about, go over the entire car & correct any obvious faults & as long as you don't find any of the really serious rot (as above), then get it in for a test NOW & see what it fails on - if anything.........then you've got a month to sort it out or whatever
