How to fix precession in pedal axle? The axle on my bike that supports the pedals appears to have loosened, so that there is precession in it - I can slightly jiggle the pedals back and forth in a perpendicular direction to the axis.

Since this is a cheap bike, I am not particularly surprised. Is it possible to correct this, for instance by buying a new set of pedals and replacing them?
You have a loose bottom bracket. It needs to be replaced.

You need to buy a new bottom bracket, along with two special tools: A crank puller and a bottom-bracket tool.

If your current bottom-bracket is cup-and-cone style (I'm betting it is), it can be removed with basic tools and elbow grease. If it's cartridge style, it can be removed with the BB tool.

Here are links to all three at Amazon:

BB

BB tool

crank puller

Or, you could take the bike to your LBS and have them handle it. This is a common operation and probably wouldn't cost much.