This engine is running rich, meaning it is putting too much fuel in for the amount of air.
The cat is unlikely to be a cause of this, but is struggling to clean up the exhaust, as it needs the gas coming out of the engine to be within quite narrow parameters.
The fuelling is controlled mainly from the input from the primary oxgen sensor. The primary sensor is before the cat, and the computer will adjust mixture values as a result of these readings. The secondary sensor, after the cat, will give a check that the cat is functioning, but will give bad readings if the input is wrong.
It is possible that a new primary sensor will fix the issue, but this should not be done until after proper diagnosis.
As said above, a check of the exhaust is necessary first, to ensure there are no leaks. A leak is unlikely, as this is checked by the MOT tester, and you;ve not reported a problem there. The MOT is just a check for safety and legality, and no diagnosis is done for that fee.
Then a computer needs connecting to monitor the readings from the sensors, preferably from a cold start, and this will hopefully show what is happening.
As said above, general engine condition will affect the efficiency of the combustion process, so a compression check, and valve clearance check should also be carried out. No parts should be changed until after a proper diagnosis. Some people have had leaking injectors, so they continue to dribble fuel after they should cut off. This is unusual, but a possibility.
