The click you're hearing is a relay. There's a relay that kills the battery saver circuit (interior lighting, radio, possibly power windows and locks) upon turning the ignition to 'start' and then resets afterwards. However, this relay does not kill absolutely everything.
In the power distribution box underneath the hood, relay number 6 is the starter relay. I would start by swapping this relay with another relay in the box itself. If the problem moves or goes away, you've found your problem.
The fuse for the ignition and starter is maxi fuse number 5. It's going to be a large (physically) 50 amp rated fuse.
Also, confirm the battery is good. take it to an auto parts store, they'll check it for free or very little, like autozone. A bad battery can cause a multitude of problems, even po300, random misfire. The battery is the beginning and end of the electricial system. Also make sure the copper wires themselves are clean and snugged in the terminals firmly.
I recommend cleaning the cables and terminals with a torch and some flux. Remove the battery before using a torch of any kind. Flux will eat any corrosion present and leave a nice clean surface. Also make sure the actual terminals themselves are clean, on the post side as well. A cleaner can be had for 5 bucks or less.
And, btw, the lead terminals like this, they suck. Like seriously, ditch these.
These are basically budget terminals. I'd recommend seeking brass versions. Brass versions are a lot better. Like this.
And, one last thing. Stay away from battery terminal protection spray. Those have an insulating quality to them, and they make everything an absolute mess.