Originally Posted by dddd

See post (?).

With SPD pedals, unless they are the cheapest versions (or with very high usage), the problem is typically the long, threaded "nut" that secures the barrel/race of the spindle cartridge solidly inside of the pedal body.
The external spline (if plastic) or the external hex (if aluminum) should first be checked for tightness.

Sometimes when the nut has been loose for a while, the contact surfaces in the pedal body may have worn to the point where the nut only bottoms out without securing the cartridge barrel/race solidly in the pedal body, but there is a fix for that.
With the spindle cartridge removed from the pedal, a few swipes with a file on the snout on the pedal body will allow the nut to tighten in further towards the barrel/race inside of the pedal body, thus allowing full tightening tension to bottom on the race instead of on the pedal body.
And with the spindle cartridge out, the bearing adjustment can be checked and corrected at the same time if needed.

The cheap SPD pedals with the plastic splined nut are somewhat notorious for their tendency to having the bearings (and/or the inside of the pedal body) wear and become loose, so periodic bearing readjustment and/or filing of the body will be necessary. They work well enough with the bearings somewhat loose, so most folks won't even notice that the bearings have excess play until the rubber seal is getting squashed and possibly severed/dislodged by excess axle movement.
It's on these cheaper SPD's that diagnosing where the looseness is coming from can take an extra step or two after a first attempt is made of just tightening the external retention nut. The cheap SPD-R road pedals, even the "carbon" R550 model, are the same in this regard, having cheap/soft bearings and plastic retention nuts.