@RestResource(urlMapping='/approve/*')
global with sharing class exAppClass {
  jstcl__TG_Timesheet__c ts = new jstcl__TG_Timesheet__c();
        // GET request
       @HttpPost
        global static String doGet() {
        RestRequest req = RestContext.request;   
          String secid = req.params.get('id');
          String approverId = req.params.get('appid');
          String ipString = req.params.get('ip');
          System.debug(secid);
          String alreadyApproved = 'expired';
          String error = 'error';
          String success = 'approved';
          String primApprover;
          String secApprover;
          jstcl__TG_Timesheet__c ts;
          ts = [SELECT Id, jstcl__Week_Ending__c, jstcl__Status__c, jstcl__Placement__r.jstcl__SecondaryApprover__c, jstcl__Placement__r.jstcl__TimecardApprover__c FROM jstcl__TG_Timesheet__c WHERE Id = :tsid FOR UPDATE];
          primApprover = String.valueOf(ts.jstcl__Placement__r.jstcl__TimecardApprover__c);
          secApprover = String.valueOf(ts.jstcl__Placement__r.jstcl__SecondaryApprover__c);  
          if (ts != null && ts.jstcl__Status__c == 'Approved' && (primApprover == approverId || secApprover == approverId )) {
            return alreadyApproved;
          } else if(ts != null && ts.jstcl__Status__c == 'Submitted' && (primApprover == approverId || secApprover == approverId)){
            try{
                insertLog(ts, approverId, ipString);
                ts.jstcl__Status__c = 'Approved';
                update ts;   
            } catch(Exception e) {
                System.debug('Exception caught: ' + e.getMessage()); 
                return error;   
            } 
          return success;
          }else {
          return error;
          }
        }
     
        private static Express_Approval_Log__c insertLog(jstcl__TG_Timesheet__c ts, Id approverId, String ip){
          Express_Approval_Log__c log = new Express_Approval_Log__c();
          log.Name = 'EAL '+ts.jstcl__Week_Ending__c;
          log.Timesheet__c = ts.Id;
          log.Approval_Date_Time__c= DateTime.now();
          log.Status__c = 'Approved';
          log.Timesheet_Approver__c = approverId;
          log.IP_Address__c = ip;
         try{
            insert log;
         }catch(Exception e) {
            System.debug('Exception caught: ' + e.getMessage());    
        } 
          return log;
      }
}

Test class so far:

@isTest
private class exAppClassTest {
    static testMethod void  updateMethodTest(){
     jstcl__TG_Timesheet__c ts = createTestRecord();
        
     String JSONMsg = '{"id" : "'+ts.Id +'","ip":"12:34:00:00","appid":"a4t3700000345ZC4AAO}';

     RestRequest req = new RestRequest();
     RestResponse res = new RestResponse();
     
     req.requestURI = '/services/apexrest/abc/xyz/';  //Request URL
     req.httpMethod = 'POST';//HTTP Request Type
     req.requestBody = Blob.valueof(JSONMsg);
     
     RestContext.request = req;
     RestContext.response= res;
     
     Test.startTest();
         exAppClass.doGet();
     Test.StopTest();   
 }
 
 // Helper method to create test data
 static jstcl__TG_Timesheet__c createTestRecord() {
     // Create test record
     Contact con=new Contact(
        FirstName='fname',
        LastName = 'lname',
        Email = 'email@gmail.com',
        Phone = '9743800309'); 
    insert con; 
    ts2__Placement__c plc = new ts2__Placement__c();
    plc.jstcl__SecondaryApprover__c = con.Id;
    plc.jstcl__TimecardApprover__c = con.Id;
    insert plc;
    jstcl__TG_Timesheet__c ts =new jstcl__TG_Timesheet__c ();
    date mydate = date.parse('12/27/2009');
    ts.jstcl__Status__c='Test';
    ts.jstcl__Week_Ending__c=mydate ;
    insert ts;
    Express_Approval_Log__c log = new Express_Approval_Log__c();
    log.Timesheet__c = ts.Id;
    log.name = 'Test';
    log.Timesheet_Approver__c = con.Id;
    log.Status__c = 'Approved';
    insert log;
     return ts ;
 }
}