The contacts are probably burned and not making good continuity. You might be able to take it apart and clean them, but most of the relays are sealed units.
It sounds like the relay needs to be replaced. These are a special type called a latching relay. When it switches on, there is no constant power applied to the coil like a regular relay. There should be a part number and brand on the side, so you can do a search on Amazon, etc to find a replacement.
Dan & Ro with two Rescued Boston "Terrorists"
2015 Tiffin Allegro RED 38QRA
Previously '06 Itasca Sunrise 35A on a W22 Workhorse
'12 Jeep Wrangler Sport and/or '08 Honda Goldwing