David P. is correct the annotations relate to applications for naturalization, but the numbers are NOT the petition numbers (they are application for a certificate of arrival numbers). These annotations have three pieces of information, two of which can help you. The three data points are the prefix number, the application number, and the date.
The first (prefix) number indicates the US naturalization district where the application was filed. Above, David Rosen posted the link to a site where one can find a table where you can translate the number into a geographic district (as of the date in the annotation).
The date annotated is the date the manifest record was checked, telling you (approximately) when their application was processed. The actual date of naturalization can be weeks, months, or even years later.
Thus annotations such as you describe and are seen on the example you shared can help one reach conclusions like "this immigrant was living in the St Louis District and trying to naturalize in 1935" or "that immigrant lived in the Boston District and was trying to naturalize in 1941."
If you do later find the naturalization petition, it should include a blank for "Certificate of Arrival No. ____ filed" and that certificate of arrival (c/a) number should match the application number annotated on the manifest. Matching those numbers tells you the US Gov't believed the person listed on the manifest and the person named on the naturalization petition were one and the same person.