Removing dissolved sugar from a recipe, as in your soup, practically impossible.
You have two main choices: Reduce the impact of the sweetness.
Increasing acidity (lemon juice or vinegar for example--since ketchup contains vinegar, vinegar or more ketchup may be most compatible with your specific recipe) may mitigate how sweet the soup seems.
This may or may not work--you would have to experiment, and it could make the soup taste worse worse (the risk being you would then still have to discard the soup.)
If you really like the borsht, and can eat (or freeze)
it all: make a second batch with considerable lower sugar, and combine them.
You will now have twice as much soup, but flavor balanced.
cooking.stackexchange.com/users/14401/SAJ14SAJ
I'm adding this answer more for completeness than anything else but it is possible to reduce the sweetness in a dish by using a sweetness inhibitor.
From McGee On Food
And Cooking : Not only are there artifical sweetners: there are also substances that block us from experiencing the sweetness of sugars...
Lactisole is phenolic compound found in small quanities in roasted coffee...
In very small amounts it reduces the apparent sweetness of sugar by two-thirds.
From Wikipedia :
At concentrations of 100–150 parts per million in food, lactisole largely suppresses the ability to perceive sweet tastes, both from sugar and from artificial sweeteners such as aspartame.
A 12% sucrose solution was perceived like a 4% sucrose solution when lactisole was added.
However, it is significantly less efficient than gymnemic acid with acesulfame potassium, sucrose, glucose and sodium saccharin.
Research found also that it has no effect on the perception of bitterness, sourness and saltiness.
According to a recent study, lactisole acts on a sweet taste receptor heteromer of the TAS1R3 sweet protein receptor in humans, but not on its rodent counterpart.
There is a branded blend of lactisole, sucrose and maltodextrin called Super Envision, which is the only commercially available version of this product afaik.
It is used in concentrations of 0.5%-1%, e.g., if you have 500g of soup then you use 2.5g-5g of Super Envison.
It can be found on websites selling Modernist cooking ingredients such as Modernist Pantry: http://www.modernistpantry.com/super-envision.html
cooking.stackexchange.com/users/7552/Stefano
I would remove some of the borsch and replace with water (removing some of the sweetness) and then rethicken it with a souring agent such as Amchoor (ground unripe Mango).
Then add soured cream at the end to further thicken, improve consistency and remove sweetness.
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I was making Spanish rice, which, called for Ketchup to taste.
Uh....the lid wasn't on tight
and I put more than I liked.
Result
....
dish was too sweet.
Not undaunted, I considered adding tomato paste, which is not usually sweet and is in fact, slightly bitter due to the concentration of tomatoes.
Added, until the sweetness was virtually gone.
A good save for me.....
Try it next time but be sure that you taste the paste and make sure it's not sweet.
cooking.stackexchange.com/users/22531/Max
I have always used pepper to reduce the taste of too much sugar and this also works the other way although sometimes I use a different sweet substance than sugar if I have put too much pepper in a sauce.
cooking.stackexchange.com/users/19464/Ian
Salt, pepper, hot sauce.
All in very judicious doses.
Worked for me
cooking.stackexchange.com/users/35957/Debi Smith
I was making this recipe:  Cabbage Borscht , which was from the old Lincoln Del, one of my mom's favourite places. I mistakenly doubled the sugar, putting in 1/2 a cup instead of a 1/4 of a cup. Given the 2/3 of a cup ketchup, which was Heinz, the soup turned out way, way too sweet.   Is there any way I can rescue this?
Too much sugar in soup -- how to make it less cloying?
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cooking.stackexchange.com/questions/29390
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Salt, pepper, vinegar, hot sauce or tomato paste may be able to mitigate the sweetness, or you could replace some of the borscht with water before thickening it again with a souring agent like Amchoor. You can add something acidic or add some pepper.
Use an acidic food to offset the sweetnes like tomato paste. Add some water and then rethicken with something sour. Add something peppery or some hot sauce.