The logic that allows episodes to be condensed into a single continuous care episode is a discharge code of 22 followed by an admission code of 4 on the same day.
Note:epn field from the source table represents each episode, but episode_c_care_key for the output is a derived field which can be a row number.
drop table #source
CREATE TABLE #source(patidid varchar(20),epn int,preadmitdate datetime,adminttime varchar(10),
admitcode varchar(10),datedischarge datetime,disctime varchar(10),disccode varchar(10))
INSERT INTO #source VALUES
(1849,1,'4/23/2020','7:29',1,'7/31/2020','9:03',22)
,(1849,2,'7/31/2020','11:00',4,'7/31/2020','12:09',22)
,(1849,3,'7/31/2020','13:10',4,'8/24/2020','10:36',10)
,(1849,4,'8/26/2020','12:25',2,null,null,null)
,(1850,1,'4/23/2020','7:33',1,'6/29/2020','7:30',22)
,(1850,2,'6/29/2020','9:35',4,'7/8/2020','10:51',7)
,(1850,3,'7/10/2020','11:51',3,'7/29/2020','9:12',7)
,(1850,4,'7/31/2020','11:00',2,'8/6/2020','10:24',22)
,(1850,5,'8/6/2020','12:26',4,null,null,null)
,(1851,1,'4/23/2020','7:35',1,'6/24/2020','13:45',22)
,(1851,2,'6/24/2020','15:06',4,'9/24/2020','15:00',2)
,(1851,3,'12/4/2020','8:59',0,null,null,null)
,(1852,1,'4/23/2020','7:37',1,'7/6/2020','11:15',20)
,(1852,2,'7/8/2020','10:56',0,'7/10/2020','11:46',2)
,(1852,3,'7/10/2020','11:47',2,'7/28/2020','13:16',22)
,(1852,4,'7/28/2020','15:17',4,'8/4/2020','11:37',22)
,(1852,5,'8/4/2020','13:40',4,'11/18/2020','15:43',2)
,(1852,6,'12/2/2020','15:23',2,null,null,null)
,(1853,1,'4/23/2020','7:40',1,'7/1/2020','8:30',22)
,(1853,2,'7/1/2020','14:57',4,'12/4/2020','12:55',7)
,(1854,1,'4/23/2020','7:44',1,'7/31/2020','13:07',20)
,(1854,2,'8/3/2020','16:30',0,'8/5/2020','9:32',2)
,(1854,3,'8/5/2020','10:34',2,'8/24/2020','8:15',22)
,(1854,4,'8/24/2020','10:33',4,'12/4/2020','7:30',22)
,(1854,5,'12/4/2020','9:13',4,null,null,null)