SQL Server FOR JSON PATH, Exclude specific field if null

I have table with data like this

Id  |  Name   | Phone  | OtherField 
----+---------+--------+-----------
 1  | ABC     | 12344  | NULL
 2  | XYZ     | NULL   | NULL

I want a SQL query to transform it like this

[
    {
      "ID":1,
      "Name":"ABC",
      "Phone":[
               {"Home":"12344"}
             ],
       "OtherFields":NULL
     },
    {
     "ID":1,
     "Name":"ABC",
     "OtherFields":NULL
    }
]

I know about INCLUDE_NULL_VALUES it includes all the empty field. I want to include all other fields except Phone.

I have edited my answer as you have changed your original request.

I don't believe you can have it both ways, keeping some NULLs and not others. The best way I can think of at the moment is to use ISNULL on columns you must keep.

For example:

DECLARE @Table TABLE ( Id INT, Name VARCHAR(10), Phone VARCHAR(10), OtherField VARCHAR(10) );
INSERT INTO @Table ( Id, Name, Phone ) VALUES
    ( 1, 'ABC', '12344' ), ( 2, 'XYZ', NULL );

SELECT 
    Id, Name, 
    JSON_QUERY ( CASE
        WHEN t.Phone IS NOT NULL THEN x.Phone
        ELSE NULL
    END ) AS Phone,
    ISNULL( OtherField, '' ) AS OtherFields
FROM @Table t
CROSS APPLY (
    SELECT ( SELECT Phone AS Home FOR JSON PATH ) AS Phone
) x
    FOR JSON PATH;

Returns

[{
    "Id": 1,
    "Name": "ABC",
    "Phone": [{
        "Home": "12344"
    }],
    "OtherFields": ""
}, {
    "Id": 2,
    "Name": "XYZ",
    "OtherFields": ""
}]