It sounds like damage from heat/moisture.
It is possible to very carefully color the spots, but it takes a lot of practice.
The more common fix for solid wood tables is to sand the finish until the top is uniform and then apply a new finish.

Much of the furniture finishes from factories are tinted to speed production.
They save a staining step.
This produces color changes if the finish is damaged though.

Before you do anything drastic, there are many common approaches to removing finish blushes. These are caused by moisture getting trapped in the finish.

Everyone seems to have a favorite. Here are frequently offered ones, in no particular order. Note that some have potential to cause unintended consequences.

Personally, I use the cloths because they have always worked for me, get immediate and visible results, and I can re-adjust the gloss level when I'm done. If I didn't have the specialized materials, I'd probably start with the mayonnaise or denatured alcohol.

Heat

Gently heat the area with a hair dryer to see if that drives out the moisture.

Mayonnaise

Smear mayonnaise on the spot, cover with plastic wrap and leave sit overnight. The next day, clean off and see if the spot is gone. It may require a second application. Some people report similar results with peanut butter or Vaseline.

Stain and ring removal cloths

Paint stores carry a yellow felt cloth that contains oils and a light abrasive. With rubbing, the oil displaces the moisture. Clean off the oil when finished. This may raise the sheen on flatter finishes.

A similar approach is baby oil and cigarette ashes, gently rubbed in.

Blush Remover / Blush Eliminator

Professional touch up technicians have an aerosol available that's lacquer thinner in a can. Spraying on a light coat dissolves the lacquer and gives the moisture a chance to evaporate away before the lacquer sets back up. This is for lacquer finishes only.

On mild cases, you may get a similar result by spraying on a light coat of aerosol lacquer, which are heavy in solvents.

Denatured alcohol

Gently pad a denatured alcohol dampened cloth over the spot in a sweeping motion. The alcohol is miscible in the water and they should evaporate away together. This is very effective on shellac finishes, but the alcohol is a solvent for shellac, so go gently. Aged lacquer is also susceptible to alcohol.

Proprietary products

Finish revivers such as Howard's Restor-a-finish can sometimes be effective at removing rings. Some of these products have methanol (an alcohol) or light mineral oil in them.

If the finish is flaking off (lost adhesion), you will have additional problems and a more extensive repair or refinishing will be in order.

Here is a bookcase that I did this year that had sat under a blanket in a leaky moving van for a week. It took about 20 minutes, including a new top coat.
Before

After

Brickeyee and Bobsmyuncle, thank you for your responses. The mayo or heat did not work, but I think I will try the cloths. If that does not work, I will probably leave it as is. Since the spots are not awful, I am afraid that I would end up making it look worse if I attempted sanding, and putting on new finish. Bobsmyuncle, the before and after pictures you posted were amazing!