You will not be able to adjust this after completing the online ID change.
To be honest, though, this sounds like it's been a complex feature for Sony to implement. It will begin as part of the PlayStation Preview Program for those who have signed up for the betas for previous PS4 software updates. If this is an issue, you can revert back to the original name for free at any time (but just once during the preview program). Subsequent changes will cost £7.99/$9.99, or £3.99/$4.99 if you have PlayStation Plus.
Finally, after months of hints and rumors, Sony has officially announced that the option to change your PSN Online ID is now being tested and will roll out to all PlayStation users in "early 2019" (with earlier testing for members of the PlayStation Preview Program). For those that do get access, you'll be able to do your first change for free. You'll also get the ability to display your previous PSN ID alongside your new one (so your friends can recognize you).
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Other nominees are Pernille Harder, Lindsey Horan, Fran Kirby, Sam Kerr, Lieke Martens, Megan Rapinoe and Christine Sinclair. Liverpool's Brazilian goalkeeper Alisson Becker, Frenchman Karim Benzema and Edinson Cavani completes the five-man list.
To make the change from xxxSniperJoe420xxx to "MyWifeIsMyLife2012" you'll merely need to head to the settings menu or profile section of the PS4. "For PlayStation Plus members, it will cost $4.99".
The preview of this feature is scheduled to end at the end of November, with a full roll-out planned for early 2019.
Another caveat is that the new feature won't be compatible with all games. When PlayStation Network online ID changes officially launch, Sony will publish a list of compatible games released before April 1, 2018. During the preview program you can revert back if the errors are so busted that you can't use programs correctly.