I might first rewrite your query using a series of joins:
SELECT
p.name,
p.owner
FROM project p
INNER JOIN
(
SELECT a.user_name
FROM users a
LEFT JOIN users b
ON a.manager = b.firstname
WHERE
a.user_name = 'sridhar.ps' OR
a.manager LIKE '%sridhar p%' OR
b.manager LIKE '%sridhar p%'
) t
ON p.owner = t.user_name
WHERE
p.customer_code = 'OTH_0071';
Then build the Laravel query using a raw subquery to represent the table aliased as t above:
$subquery = "(SELECT a.user_name FROM users a LEFT JOIN users b ";
$subquery .= "ON a.manager = b.firstname ";
$subquery .= "WHERE a.user_name = 'sridhar.ps' OR a.manager LIKE '%sridhar p%' ";
$subquery .= "OR b.manager LIKE '%sridhar p%') AS t";
DB:table('project)
->select('name', 'owner')
->join(DB::raw($subquery), 'p.owner', '=', 't.user_name')
->where('customer_code','OTH_0071')
->get();
This join approach might be more performant than what you currently have. In any case, you can test this answer, compare it against the answer by @Gaurav, and then use whichever works the best.