In order to convince your employer to give you that time off, you will need to present a case that it is in someway beneficial to them, or at least that they are not any worse off for it. In general your full-time employer will see no benefit to you working for another company instead of them when you have already agreed to a full time contract.
Since you have mentioned your company don't know about your second job, it's an awkward proposition to make. Some companies are flexible, but there is no way you will be able to convince them to work for your second employer on Friday without having it affect your schedule. In a best case scenario if your employer is willing to work with you, you will need to make up that time somewhere else. Remember, what you want does not matter to your employer. They hired you (and you agreed) to work full time and from their perspective you're asking if you can work less for no benefit to them.
Since you say you don't think your employer would take it well, I would instead try to work with your second employer to see if they're fine with you working some other hours. If not, you'll probably have to make a choice as to which job you like more. You can't always have it both ways.