Hello!
I am new to this forum and I’m hoping I can get some help.
I purchased my bonsai ume (peach) tree in the spring.
After the flowers bloomed, the leaves seemed to always be dry, droopy, and curled inwards with burned tips as shown in the pictures.
I’ve also noticed some white spots on the branches, could this be some kind of fungus? If so, how would I treat this?
Any advice would be greatly appreciated. Thanks in advance.

The soil looks very dry. There is scale insects on the branches which is not helping, use a cotton but and metylated spirit to get rid, or a mild soapy solution but very weak.

Should I remove all the leaves too or repot it?

Absolutely NOT Do not remove leaves do not re pot it. Spring is the only time to re pot and this plant is stressed. Water it. Spray it and see how it goes.

DUring summer, prunus mume rolls its leaves on those spots where the next year a flower will develop instead of a side-branch. It is a little dry maybe, but else.. Congratulations. Your ume will flower in spring. This is normal. Prunus mume has naturally white spots on branches.