How do you fix gaps in the bathroom?

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You can certainly re-grout the area and you should find these step-by-step articles helpful: How to remove grout and How to grout wall tiles.

However, if there has been some movement I would suggest using a flexible silicone as it is more forgiving in such a scenario. Begin by removing any existing grout that is loose with a DTA Tile Scriber And Grout Remover. You should then find these step-by-step articles of assistance: How to apply bathroom sealant and How to silicone a gap.

An appropriate product to use would be Selleys 300g Wet Area Waterproof Silicone Sealant - White. - MitchellMc