[PROBLEM] No internet access via homeplug

[sassafras]

Is it possible the network signal is being weakened or corrupted in the mains wiring?

As the homeplug relies on filtering the network signal from the mains, perhaps there are appliances on the offending power circuits which are causing interference. You could try turning off as many appliances as possible and see if that helps.
Alternatively the problematic outlets may have too much wiring between them and the source, thus a weak signal.

The laptop may work as it may be able to resolve the signal from the noise more effectively.

Just a thought.