Hi Robin,

You might try looking at old city directories for Plainview, Long Island.  The local library may have old copies on microfilm. 
https://poblib.org/index.php?option=com_dropfiles&task=onedrive.onedrivesync
I haven't found one on-line  (including Ancestry.)  Maybe one is online and I just couldn't locate it.  If someone knows of one on-line can they please post a link?

Old city directories usually have listings by type of business.  Reformatories were sometimes called "training schools."

I am assuming that they lived in Manhattan.  NYC Municipal Records has:
https://www.familysearch.org/en/wiki/New_York_City_Department_of_Records
"The criminal justice records 1684-1966 are the largest and most comprehensive collection of their type in the English-speaking world."

There may be a card regarding his arrest / trial.  Many years ago I found there my uncle's Sing Sing admission card from the 1910's.

The 1908 American Jewish Yearbook may list the reformatory or training school name, if it was a Jewish run institution.
http://www.ajcarchives.org/main.php?GroupingId=10040

If he was still residing at the reformatory in 1910 than you should find him listed in the 1910 Federal Census as an inmate.  Try searching his name.
https://www.familysearch.org/search/collection/1727033