The procedure of how to fix a loose fuel pump relay involves changing a connector assembly with the wires. This is known as a pigtail. This is because there are multiple wires connected to a fuel pump relay, the only way it can come loose is if the connectors on those wire terminals are bad. The reason why you need to change the entire pigtail is because the ends that push onto the terminals on the relay are what mess up. They get loose in the plastic body of the connector that holds them and can even fall out leaving you with a very hard to find (and frustrating) wiring problem. Another (free of charge) way you can do it is to remove the connector, and put the wires back on the correct terminals on the relay. This is not recommended because it makes future diagnosis and repairs to this system more difficult.
Since the correct way to fix a loose relay is to change the pigtail. Changing a pigtail is not hard to do, as it only involves removing the old connector by cutting the wires (1 by 1), and splicing them on the correct corresponding wire on the replacement part. Once each wire has been cut loose from the old connector, and the new connector (pigtail) spliced back into place, the repair is done. As far as how to splice the wires together, there are many ways to do this, and information on how to splice wires together can easily be found with a quick search in the search bar at the top of this page. The preferred way is to solder the wires together, and cover them with heat shrink, but many people do many different things, and it really does not matter as long as a good connection, that will not come loose or let naked wires touch anything, is made in the splice.
Of course all of this assumes that you already know that the problem is with the fuel pump relay connector. Finding the cause of problems like this is usually a daunting task, and people who go this far are usually just changing parts and hoping for the best results. When things don’t always work out the way you wanted them to, you need to get some help. Knowing how to diagnose electrical problems and how to use a voltmeter would be essential skills needed in order to properly diagnose a problem like this. Though it is common to have bad relay connectors on some GM products from the early 2000s, simply pushing the loose terminal back into the connector will fix it. This is also the with those pesky MAP sensor connectors that cause so many engine lights to come on in the 90s model vehicles. Mechanics are happy that the connectors were redesigned to something that is a lot sturdier.
Thanks for reading and good luck!