If you can, try to pair him with a munna. Not only are munna extremely non-threatening pokémon (that is to say, they’re just plain calming to look at), but they also consume nightmares as their basic form of sustenance. As a fellow psychic-type, it will likely do everything it can to comfort your espurr, especially if the munna is older.
However, if you don’t have access to a munna, encourage your meowstic to lie down with their kitten and comfort him as he sleeps, at least for the next couple of months. Eventually, try to have him “weaned” off this care by letting his parents lie with him only until he falls asleep.
It may also help to reduce stress through aromatherapy. In fact, the various incenses were originally created for the express purpose of lowering a pokémon’s stress and heightening their mental serenity and clarity. (The secondary effects of each type were simply perks that humans had stumbled across sometime afterward.) Of the various types, psychics tend to respond most to odd incense, but pure incense is just as calming … and likely less aggravating to the human nose. Once you’ve chosen a scent that your espurr likes (and you wouldn’t mind filling your home or apartment), place it in its special holder and place the activated holder somewhere near your espurr, in a spot where its scent can’t be blocked and where the holder can’t be knocked over by his developing telekinetic abilities.
Of course, on top of all of this, see if there are any possible causes of stress in your espurr’s waking life. Considering he was only recently born, it’s doubtful, but nonetheless, it’s entirely possible that your espurr may have been worked up over something that didn’t quite get resolved before he was put to bed. Also check for any possible ghost-types or members of the drowzee line in the vicinity, as these could cause nightmares as well.
The Pokédex According to Bill 