No, not if the socket is on the ring and, if you conceal the cable <50mm deep it should be RCD protected or earth screened cable. These cable should be run vertically or horizontally from accessories.
How powerful is the fire? Generally you would use a dedicated circuit for fixed appliances over 2kw,eg immersion heater and oven.
For example if you have that fire and an immersion on the go, you'd only have 6A left for everything else.
Have a think about what you have on that circuit.
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