They look like Pansies.. so no, don't cut them back. Do deadhead them though (cut the flowers off when they've died down). Pansies wont re-flower once they've produced a seed head.
Take the dead flower/seed pod and its stalk back as far down as you can. If you just take off the flower/pod it's better than doing nothing, but the old stalk won't do anything but look untidy.
Doncaster, South Yorkshire. Soil type: sandy, well-drained
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