Exhaust emissions are tested to a set sequence and procedure, involving engine revs at particular levels. Fast idle is in the range of 2500-3000rpm, normal idle is in the range of 450-1500rpm.
The lambda sensor is fitted in the exhaust and sends signals to the engine management system to adjust fuelling, amongst other things. If yours is faulty it may be causing the engine to run too rich, as indicated by the excessive emissions.
Your re-test will only be free if the vehicle is left with the original MOT testing station for repair and then a partial re-test is carried out within 10 working days of the original MOT test.
So if you take the vehicle away and either fix it yourself or get another garage to fix it you will need to bring it back for a partial re-test within 10 working days to get the partial re-test fee which is half the full test fee.
If you bring it back to the original MOT testing station outside of 10 working days then the station will have to perform a complete MOT test and will charge the full MOT fee.
If you leave your vehicle with your MOT testing station to repair and re-test within 10 working days then only a partial retest will be necessary and this will be free.
A partial retest is where only the items that you vehicle failed on during the full MOT test are examined.
If your vehicle fails on the partial re-test then the vehicle will have to have a full MOT test when you bring it back again and the full fee will be charged.