MySql - Remove duplicate record if another column is empty

I think this is what you want...

I assume your table is called t:

 CREATE TABLE t
 (
     `ID` int PRIMARY KEY, 
     `First_Name` varchar(50), 
     `Last_Name` varchar(50), 
     `City` varchar(50)
 ) ;

 INSERT INTO t
     (`ID`, `First_Name`, `Last_Name`, `City`)
 VALUES
     (1, 'Alan', 'Smith', 'Los Angeles'),
     (2, 'Alan', 'Smith', NULL),
     (3, 'Alan', 'Smith', 'New York'),
     (4, 'Alan', 'Smith', NULL),
     (5, 'Alan', 'Smith', 'Houston')
 ;

This query DELETEs the rows you don't want...

 DELETE
 FROM
     t
 WHERE
     t.id IN
     (
     SELECT 
         *
     FROM
         (
         SELECT
             id
         FROM
             t AS t1
         WHERE
             City IS NULL
             AND
             EXISTS
             (
                  SELECT 1
                    FROM t AS t2
                   WHERE     t2.`First_Name` = t1.`First_Name`
                         AND t2.`Last_Name`  = t1.`Last_Name`
                         AND t2.`City` IS NOT NULL
                         AND t2.`ID` <> t1.`ID`
             ) 
         ) AS s0 
     ) ; 

 SELECT * FROM t

 ID | First_Name | Last_Name | City

 -: | :--------- | :-------- | :----------
  1 | Alan       | Smith     | Los Angeles
  3 | Alan       | Smith     | New York
  5 | Alan       | Smith     | Houston
 

dbfiddle here

--

NOTEs:

1. The need of a subquery within of a subquery is a limitation of MySQL (most SQL databases don't need it). See You can't specify target table 'NAME' for update in FROM clause

2. I've assumed that you wouldn't want to delete any row if there is not one with a City. I.e.: if you have (15, 'John', 'Doe', NULL) and (16, 'John', 'Doe', NULL) you leave both on the table.