Hi and welcome to the forum
Your mum's tank has a high nitrite reading and that is probably why the glowlight tetra is dying. The tail might have been damaged when it was sucked up against the filter intake. The fish got sucked onto the filter intake because it is being poisoned and may not live.
Your tank has an ammonia reading, which would suggest the filter hasn't been cycled properly, or you are over feeding, or you replace filter media when you clean the filter.
You can do large water changes as long as the new water is free of chlorine/ chloramine, and it has a similar chemistry (GH, KH & pH) and temperature to that tank water. The temperature doesn't have to be exactly the same and it's fine if it is a couple of degrees cooler or warmer. But the pH, GH & KH should be similar or the same as the tank. If you do regular water changes (once a week) then the tank water chemistry should be similar to the water supply.
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AMMONIA & NITRITE LEVELS
You need to keep ammonia and nitrite levels on 0ppm at all times, and nitrates below 20ppm. If you ever get an ammonia or nitrite reading above 0ppm, do a 75% water change and gravel clean the substrate every day until the levels are 0ppm. The fish are more likely to die from ammonia or nitrite poisoning than from a big water change.
Make sure any new water is free of chlorine/ chloramine before it's added to the tank.
If you have concerns about the river water's temperature, fill up some buckets and bring them inside. Leave them to sit near the tank for 24 hours and they will warm up to room temperature. That should get the water temperature closer to the tank's temperature.
You can also boil some tap/ river water and add some boiled water to a bucket of water to raise the temperature.
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FILTERS
What sort of filters do you have?
How often and how do you clean the filters?
If you have filters that contain filter pads/ cartridges, do not replace these pads unless they are falling apart. Companies that sell these types of filter are stealing your money by telling you to replace the filter pads every month. That is bad for the fish because you get rid of the beneficial filter bacteria living in and on the filter media/ materials and that bacteria keeps the ammonia and nitrite levels on 0ppm. If you replace the pads with new ones, you get rid of the good bacteria and you get ammonia and or nitrite readings that can kill the fish.
If you have filter pads, buy a sponge for a different brand of filter and use a pair of scissors to cut the sponge to fit in your filter. Sponges last for 10+ years and only need replacing when they start to fall apart. I use AquaClear sponges but there are other brands that work just as well.
To clean sponges, simply get a bucket of water from the aquarium and squeeze the sponge out in the bucket of water. If the sponge is really dirty, squeeze it out in a second bucket of tank water. When the sponge is clean, put it back in the filter and tip the bucket of dirty water on the lawn.
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WATER DILUTION RATIOS
Fish live in a soup of microscopic organisms including bacteria, fungus, viruses, protozoans, worms, flukes and various other things that make your skin crawl. Doing a big water change and gravel cleaning the substrate on a regular basis will dilute these organisms and reduce their numbers in the water, thus making it a safer and healthier environment for the fish.
If you do a 25% water change each week you leave behind 75% of the bad stuff in the water.
If you do a 50% water change each week you leave behind 50% of the bad stuff in the water.
If you do a 75% water change each week you leave behind 25% of the bad stuff in the water.
If you have a water quality issue, you need to fix it asap and the best way is with big daily water changes. In extreme cases you can change 90-100% of the water.
