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Mist them with an ironing-type spray bottle. Or use a spray wand with a head like this:
https://www.amazon.com/Haws-Small-Extra-Spray-Plastic/dp/B004R4STUI/ref=pd_day0_hl_201_6/147-4530596-9991260?_encoding=UTF8&pd_rd_i=B004R4STUI&pd_rd_r=fde78846-4791-11e9-b40f-e948c6584ac1&pd_rd_w=9s6TC&pd_rd_wg=t7dl8&pf_rd_p=ad07871c-e646-4161-82c7-5ed0d4c85b07&pf_rd_r=YEN47ZKBBV1VVWGA79SG&psc=1&refRID=YEN47ZKBBV1VVWGA79SG

Fortunately, this is an easy test to do yourself! Some plants have a sensitivity to it, and some prefer to have it to germinate.

The best way to test fluoridated vs. non is to do comparisons side by side. Many seedlings will prefer non-fluoridated water, but many mature plants will need it, at least, occasionally to prevent chlorine and fluorine deficiencies.

Depending on the plant, it may be experiencing a chlorine deficiency. Many plants benefit from tap water, as many nutrient supplements are not full supplementation. They may lack certain elements like chlorine or fluorine, which plants use both in VERY SMALL quantities.

While it is true that certain plants have a sensitivity to it, there are also many plants that need it.

What is the plant in question? Maybe we can work through the reasons that this could have happened.

I have been growing for 30 years and haven't had trouble with city tap water. Most cities have switched from chlorine to chloramine. Check with your water department. I think it is more of an issue with houseplants that are repotted every few years and where chlorine levels can build. Chlorine is highly volatile and "off gasses". Leaving tap water uncovered overnight reduces the chlorine content significantly. Chloramine does not which makes it a safer chemical to use from a safety standpoint. If you are concerned, using a filter pitcher will remove chlorine.

Perhaps a rain barrel at the garden site would be of help.

https://www.gardeningknowhow.com/garden-how-to/watering/collecting-rainwater.htm

Garden wagons can make hauling buckets of water easier; heavy duty wagons can be found at farm stores.
Recycle a child's wagon for this purpose.

The best thing to do would be to drill out holes in the bottom of the planter to allow for water to leave it faster. Good drainage is the only way here. I would also recommend using succulent soil, as it will dry out much faster. This soil will be a little too heavy for their requirements, although they will do just fine, as long as you can keep the moisture away.