Most larger supermarkets accept carrier bags as well as other plastic films. As a general rule, if you can stretch the film then it can be recycled at supermarkets (even if it is not supermarket product packaging or is labelled as non-recyclable plastic - bubble wrap as an example), except for compostable and biodegradable bags. More info and the list of plastics recyclable at supermarkets here:
Besides stretchy plastics, some M&S stores have started accepting even more types of plastics for recycling, like black ready meal trays, crisp packets, sauce sachets and certain cosmetics containers, not restricted to M&S products but from any retailers. This scheme is expected to be rolled out to all M&S stores by the end of 2019. For more info, see this and this.