"Prejudice" is a strong (could be an incorrect or biased) opinion held by a person. "To reason with" is the act of saying something in hopes the person will change their mind.
Therefore, the sentence can mean simply:" Never try say something to change the person's mind, if the person has already has a strong opinion about it. If the person cannot change his own opinion for the better, then you will not succeed either".
The second question "I can't seem to be through with the sentence on my own." I think it would be better if you say:" I can't think through this sentence on my own."