A laminating machine and ten A4 pouches costs less than 20 GBP at online shopping sites. You have the convenience, and 10 cue cards will cost 2 GBP each, comparable with High Street office suppliers' laminating charges. The more you make, the cheaper per card it becomes because you already have the machine.
But if price is the main consideration, you can cover the cue card with self-adhesive film intended for covering books. The film is a lamination layer, and laminated materials are much stiffer than the sum of their layers, when joined by adhesive.
I found book covering film on shopping sites that might cost 1 GBP per A4 cue card, half that if you cover one side only. It won't be as stiff, and may curl, but you can try that first.
Print on say 160 gsm card, which is about twice the usual thickness of paper.
Cover both sides with the film.
Costs about 1 GBP per A4 per card, half that if you cover one side.
If the cards are smaller than A4, paper of 100 gsm may be enough, with the additional strengthening of the film.