dont really do a lot of this type of work. The tap is leaking pretty bad, guessing just removing the bell shaped cover will expose the tap and then hopefully file flat and replace the washer?
Put the plug in the bath as well (saves you having to strip the trap when you drop the teeny screw holding the taphead on) and a towel in the bottom of the bath stops you damaging the enamel when you drop your spanner
I take it it is a screw down tap, not 1/4 turn? if so then it wont be a cartridge, it'll just be a 1/2" rubber washer but you get different types depending on the jumper type so you want to take it out to take with you to match it up. Do remember it'll be well over compressed so the new washer may look nothing like the old one but the supplier should be able to advise.
you can buy packs of black rubber washers in 3/8, 1/2 an 3/4, an with both flat an beveled surface washers.all incl in one cheap pack.
but first, as above, remove the old washers.
after pullin the valves shine a light in to examine the state of the tap seats.
the taps are badged wrong - the hot should be on the left but maybe the valves been piped wrong, chesck it out?
it might be worth takin off the cover plate an seein whats goin on behind it - thers probly a small allen/grub screw underneath the spout collar.
And... always check bath (shower trays) for any existing damage, prior to taking your tools in there and photograph/point out to customer. I have been wrongly accused of damage before and a colleague once spent a couple of hours magic-ing away a blemish he thought he'd caused only to be advised by the, extremely grateful, customer how thankful they were for the free repair to existing damage on their bath
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