The 16th (Service) Battalion (Public Schools), Middlesex Regiment were not Public Works Pioneers.
The Public Works Pioneers were the 18th, 19th (Service) Battalions and 25th (Reserve) Battalion. The PWPB did have a bespoke badge with an extra scroll, however, until last year there were no period photographs of it in wear by any members of any of the battalions.
Thanks to Brian (FoT) who rediscovered a picture of the PWPB badge in wear by the band of the 25th Bn:
https://www.britishbadgeforum.com/fo...ad.php?t=79061
The Battalion’s title in the annotation suggests a date of pre September 1916 and even then not ever man wears the badge. So it seem likely it was a bandsmen’s badge but not universally worn.
Back to your man, if he’s 16th Battalion or was not a musician in the 18th, 19th or 25th Bns then he will have worn the standard Middlesex cap badge.