I'll write in bold to differentiate my words from the code.
Thank you so much! With every answer I understand a bit more (I think....).

So, let me be more clear (and sorry I wasn't before. I wasn't sure what information you'd need).

The only (non test) class I changed is this:

public with sharing class TestUtilities {
    
    public Contact      aContac         {get; set;}
    public Attachment   aAttachment     {get; set;}
    
    static User testUser;
    
    public void generateContact(){

        this.aContac                = new Contact();
        this.aContac.firstName      = 'Test';
        this.aContac.LastName       = 'Test';
        
        insert this.aContac;

        this.aAttachment = new Attachment();
        this.aAttachment.Body = Blob.valueOf('String');
    }
    
    public static TestUtilities generateTest(){
        TestUtilities e = new TestUtilities();
        e.generateContact();
        return e;
    }
    
        public static User createUser(Id profileId)
        {   
            //fetch existing users with provided profile and role
            List<User> userLst = [Select Id From User where ProfileId =: profileId and IsActive = :true Limit 1];
            
            User user;
            
            if(userLst.size()>0)
            {
                //at least one user exists; return user record
                user =  userLst[0];
            }
            else
            {
                //user does not exist; create new user
                user = new User(profileid = profileId,
                                alias = 'standt', email = 'standarduser@testorg.com',
                                emailencodingkey = 'UTF-8', lastname = 'Tsting', languagelocalekey = 'en_US',
                                localesidkey = 'en_US', timezonesidkey = 'America/Los_Angeles',
                                username = profileId + '@testorg.com');
                insert user;
            }
           
            return user;
        }
        
    public static Clients__c insertClient()
    {
        Clients__c c = new Clients__c(Name ='TestClient', Active__c = true,
                                      OC_Programme__c = 'OC Central', Current_Address__c = 'N/A',
                                      Primary_Phone_Number__c = 'N/A');
        return c;
    }
    
    public static Contact insertContact()
    {
        Contact c = new Contact(FirstName ='Test',LastName ='Test', Training_Contact__c = true);
        return c;
    }
    
    public static Project__c insertProject()
    {
        Profile p = [select id from profile where name='OC Impact'];
        UserRole r = [select id from UserRole where name='OC Impact Director'];
        User userSRO  = TestUtilities.createUser(p.Id);
        RecordType rt = [select id from RecordType where SobjectType = 'Project__c' and DeveloperName = 'Project' limit 1];
        string rId = rt.Id;
        Project__c proj = new Project__c(Name = 'testProject',
                                         OC_field__c = 'OC Impact',
                                         Project_summary__c = 'Test Summary',
                                         Project_start_date__c = date.newInstance(2013, 1, 1),
                                         Project_end_date__c = date.newInstance(2013, 12, 25),
                                         RecordtypeId = rt.id,
                                         Submit_for_Approval_by__c = date.newInstance(2018, 12, 25),
                                         Senior_Responsible_Officer_SRO__c = userSRO.Id);
        return proj;                                 
    }
    
    public static Project_Budget__c createProjectBudget(Id projId)
    {
        Project_Budget__c pb = new Project_Budget__c(Name = 'Session',
                                                     Cost_Centre__c = 'Session',
                                                     Income__c = 0,
                                                     Expense__c = 0,
                                                     Project__c = projId);
        return pb;
    }
    
    public static Project_Budget_Detail__c createProjectBudgetDetail(Id projBudgetId)
    {
        Project_Budget_Detail__c pb = new Project_Budget_Detail__c(
                                                     Paid_As__c = 'Cash',
                                                     Income__c = 0,
                                                     Expense__c = 0,
                                                     Project_Budget__c = projBudgetId);
        return pb;
    }
}

I also changed 2 test clases:

1.

/**

 */
@isTest
private class AttendeeControllerTEST {

    static testMethod void testAddClientToSession() {
        
        Clients__c c = TestUtilities.insertClient();
        insert c;
        Project__c p = TestUtilities.insertProject();
        insert p;
        Event e = new Event(Subject='Test',ActivityDateTime=datetime.now(),WhatId=p.Id,DurationInMinutes=10);
        insert e;
        
        Test.setCurrentPageReference(Page.EventAttendees);
        
        AttendeeController con = new AttendeeController();
        apexPages.currentPage().getParameters().put('EventID', e.Id);

        system.assert(con.ClientList.size()==1);
        con.clientList[0].isSelected = true;
        con.SaveAttendees();

    }
    
    static testMethod void testAddClientToAllSessions() {
        
        Clients__c c = TestUtilities.insertClient();
        insert c;
        Project__c p = TestUtilities.insertProject();
        insert p;
        Event e = new Event(Subject='Test',ActivityDateTime=datetime.now(),WhatId=p.Id,DurationInMinutes=10);
        insert e;
        
        Test.setCurrentPageReference(Page.EventAttendees);
        
        AttendeeController con = new AttendeeController();
        apexPages.currentPage().getParameters().put('ProjectId', p.Id);

        system.assert(con.ClientList.size()==1);
        con.clientList[0].isSelected = true;
        con.SaveAttendees();

    }
}

2.
@isTest
private class TESTCLASS_EventTrigger {
static testMethod void validateEventTrigger() {

    string mySubject = 'Test Class Call';
    datetime myDateTime = datetime.newInstance(2012, 1, 1);
    integer myDuration = 30;
   
Clients__c c = TestUtilities.insertClient();
    insert c;
Event e = new Event(Subject=mySubject, WhatID = c.id,
                    DurationInMinutes=myDuration, StartDateTime=myDateTime);
       insert e;

// Retrieve the new Event
e = [SELECT ID, Calendar_Title__c FROM Event WHERE Subject = :mySubject];
System.debug('Calendar_Title__c: ' + e.Calendar_Title__c);
    

// Test that the trigger correctly updated the price
System.assertEquals(c.Name + ': ' + mySubject, e.Calendar_Title__c);
}
}


I think the problem is here: TestUtilities as it only has 8% coverage:







But I wouldn't have a clue on where to start to increase coverage. It may be something simple that you can help me change. I hope so, anyway :-)

Again, thank you very much.