Starter clicks using the key, engine starts by jumpering the relay

Last time one of my cars did this it turned out to be a bad (but looked good) ground cable between the body and the engine block.

You can check by using jumper cables between the neg post and any bolt you can get on on the block, head or an accessory mount.

It took me a while to figure this one out as it sure seemed like a dead battery or bad starter or relay.