I assume you just want to keep DISTRICT_NUM and not all the attributes from your Polygon layer.
Join features are written to the output feature class. The target features and the joined attributes from the Joins attributes from one feature to another based on the spatial Sounds like a Spatial Join would do the job.
That aside I have every license level and nearly every extension for ArcGIS Pro and ArcGIS Desktop. Nor can I add additional fields to either feature class. Some limitations: I don't have anyway of adding new feature classes to the overall GIS, so processes that create new output feature classes will not work. What tool or combination of tools could I use to automatically fill the DISTRICT attribute in the point feature classes based upon the DISTRICT_NUM attribute of the polygon those features lie entirely within? However, this DISTRICT attribute has many null or incorrect values, and will have to be redone in it's entirety. The point features all have an attribute (DISTRICT) that is intended to identify the district in which they lay, and this is supposed to correspond directly to the district name attribute (DISTRICT_NUM) in the district polygon feature class. In addition, I have a polygon feature class that divides the project area into distinct districts. I'm doing some keyboard-monkey level cleanup of a GIS with a vast number of point feature classes, some 50,000 points in total.