Inadequate Representation

If someone has to use a public defender because they cannot afford one when they are to appear in court, can they request a retrial or another hearing if they feel the public defender did not represent the defendant's best interest where they could use a different attorney?

Depends on what you mean by "best interest not represented". If you don't like the outcome of the trial you cannot simply get a big DO OVER, so to speak. But if you can PROVE that some constitutional violation occurs you may be able to get the ruling vacated BUT in order to do so, it's going to take a lot of time and money to go through those steps. A LOT.