The one at the bottom of the image is C1754 (0.1MF/4V/01005) and the top one is C1739 (0.01MF/6.3V/01005).
Are they necessary? You should be able to test your repair without them but if you are charging a customer for this repair, then you should replace them. Engineers put them there for a reason. They are easy to source (Digi-Key) and inexpensive.
These capacitors are used to filter the power rails from any "noise" on the line and they are put as close to the IC as possible. Considering that the Tristar IC is used for communications, I would say it is important to have them there.
If you are doing micro-soldering repairs, you really should have schematics and boardviews? There are free versions on the web or you could license ZXW Tools for less than 100$.
As an aside, you ask lots of questions and many repair techs invest time to give you thoughtful answers...yet you rarely upvote or select their answers. This isn't about just "points". It helps the many people who may find your question in the future because they have a similar problem. Seeing votes and selections gives them clues about which answers are helpful. Maybe the mods (@mayer @oldturkey03) should chime in here, I don't want to be out of line.
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