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Here is an article that you may find helpful: https://www.gardeningknowhow.com/ornamental/trees/tgen/tree-sucker-removal-and-tree-sucker-control.htm

These trees self-pollinate, containing both male and female flowers and are not either a male or female tree.

All information I found states that this is a small tree, assuming that, your tree is probably mature. Before cutting any branches, check for signs of life. It is possible the tree is just now showing it’s reaction to the hurricane. Did it get any saltwater? If so, this could cause some short-term damage. If you feel the limbs are dead, remove them properly. While it is normal for all fruit trees to drop blooms and immature fruit, it can be a sign of stress in the form of heat/cold, too much/little water and/or fertilizer.

When you say first year, I'm assuming that means the fruit that was on the tree when you bought it. If you give your tree the care it needs, and are patient, it should come around. Remember, it was removed from its nice cushy nursery to a big new world. It has to develop roots and do a bunch of growing. Other than youth, hard bitter fruit is most often caused by insufficient sun, environmental extremes, or water stress (too much then too little.) This article has more information: https://www.gardeningknowhow.com/edible/fruits/tangerine/mandarin-orange-tree-care.htm

It's best to cut out the dead branches and see if the tree recovers on its own.

If you suspect the tree might be diseased, though, you might want to remove it to prevent any disease from spreading to nearby related trees (other stone fruit trees). Make sure to sterilize your tools with a 10% bleach solution after you prune the tree to kill any potential disease organisms. Or, consider talking to an arborist or local extension agent to see if the tree has any disease problems.

Also, see these articles about plum tree diseases:
https://www.gardeningknowhow.com/edible/fruits/plum/plum-tree-diseases.htm
https://extension.psu.edu/plants/tree-fruit/diseases/stone-fruit-diseases

Sometimes, when these lack sulfur in the soil it can affect the taste. Sulfur comes in many forms. Iron sulfate will help in this case.

Here is an article for information on the care of these trees: https://www.gardeningknowhow.com/edible/fruits/plum/satsuma-plum-care.htm