Minimum standards for airline operations are set by the International Civil Aviation Organisation in Annex 6 to the Chicago Convention. It is the responsibility of individual contracting states to implement those standards for airlines which have their principal place of business within their territory (the registration of the aircraft is irrelevant). JAR-OPS in the JAA's harmonised requirement for commercial air transportation and implements Annex 6. There is absolutely no requirement for non JAA airlines to apply with JAR-OPS.
Although MK appears to base some aircraft in the UK and have some offices here I understand that it also has offices in Ghana and conducts a significant proportion of it to/from Ghana. In issuing MK with an AOC the Ghana CAA are obviously satisfied that MK has its principal place of business in Ghana. In issuing a permit for MK to conduct services to/from the UK the Department for Transport must also be satisfied that MK has its principal place of business in Ghana.
My understanding is that the CAA inspect foreign airlines at the request of DfT and it is DfT that follow up any finding. In my experience they follow these up with both the airline and the relevant national aviation authority. I do not doubt that MK have been inspected.
I am not sure what "dirty goods" have been referred to but exemptions for dangerous goods are issued by the CAA not the Secretary of State.