Hi,


Trigger:
-------------------
trigger ProjectTrigger on Project__c (before Insert) {
    if(Trigger.isBefore) {
        if(Trigger.isInsert) {
            ProjectHandler.onBeforeSave(Trigger.New);
        }
    }
}


Apex class:
--------------------------

public with sharing class ProjectHandler {
    public static void onBeforeSave(List<Project__c> projects) {
        
        List<Week__c> newWeeks = new List<Week__c>();
        
        // Loop to iterate over the list of Project records
        for(Project__c project : projects) {
            
            //Date that accepts the Start Date
            Date startDate =  project.Start_Date__c ;
            System.debug('Date of Start : ' + startDate);
            
            // It will get the start date of the week based on Date so here it is 2020 02 09
            StartDate = StartDate.toStartofWeek();
            System.debug('Start of week for Start Date : ' + startDate);
            
            Date endDate = project.End_Date__c;
            System.debug('Date of EndDate : ' + EndDate);
            
            // It will get the start date of the week based on Date so here it is 2020 02 09
            endDate = endDate.toStartofWeek() + 7;
            System.debug('Start of week for end Date : ' + endDate);
            
            while(startDate < endDate) {
                Week__c week = new Week__c();
                
                DateTime dt = Datetime.newInstanceGmt(startDate.year(), startDate.month(), startDate.day()) ;
                week.Name = String.valueOf(startDate.day())  + dt.format('MMM');
                System.debug(week.Name);
                startDate = startDate+7;
                newWeeks.add(week);
            }
            
        }
        
        try {
            if(!newWeeks.isEmpty()) {
                insert newWeeks;
            }
            
        }catch(DMLException e) {
            System.debug('Unable to insert the weeks : ' + e.getMessage());
        }
    }
}

Input:
-----------------------
Start Date:
Feb 10, 2020
End Date:
Feb 27, 2020

Output: 
---------------------
9Feb
16Feb
23Feb

I hope this helps.

Thanks,