There is a slight difference in the literal meanings, although they're used the same way. When you "work up" courage, you create it within yourself. But when you "summon up" courage, it's already within you and you only have to bring it to the surface. I would recommend: I worked up the courage to ask for a pay raise.

You work up /summon the courage to do something. I’d use ‘summon’ for something bigger. It’s a little more formal maybe. He summoned the courage to go into battle. I worked up the courage to ask her out.

For me, to "work up the courage" and to "summon the courage" (without "up") essentially mean the same thing. (The article "the" feels necessary to me too.) The only subtle difference to me is that "working up" implies a process that may have taken some time. So both of these sound good to me: * I worked up the courage and asked for a pay raise. * I summoned the courage and asked for a pay raise.