This step is executing the sp_cleanup_log_shipping_history system stored procedure.
From the docs:
sp_cleanup_log_shipping_history must be run from the master database on any log shipping server. This stored procedure cleans up local and remote copies of log_shipping_monitor_history_detail and log_shipping_monitor_error_detail based on history retention period.
Note that even though the docs specify that the cleanup procedure must be run from the master database, the two tables being cleaned up live in the msdb database. The clustering key on both of these tables is agent_id, agent_type, which are the same two parameters on the cleanup procedure, and both have a non-clustered index on log_time_utc. Thus, we can expect that they are decently indexed for cleanup.
The specific code path is that sys.sp_cleanup_log_shipping_history calls sys.sp_MSprocesslogshippingretentioncleanup (located in the resource database, and thus a little tougher to inspect), which eventually runs this code:
-- cleanup history_detail
delete from msdb.dbo.log_shipping_monitor_history_detail
where agent_id = @agent_id
and agent_type = @agent_type
and log_time_utc < @cutoff_time_utc
-- cleanup error_detail
delete from msdb.dbo.log_shipping_monitor_error_detail
where agent_id = @agent_id
and agent_type = @agent_type
and log_time_utc < @cutoff_time_utc
Given the intermittent nature of the failures, I suspect that you are encountering some sort of blocking that prevents the cleanup from running normally. The most likely scenario is when you have multiple log shipping jobs running in parallel, and they are concurrently trying to write to the same table.
To know the reason for sure, you would need to reproduce the issue, or catch it with monitoring, or catch it in the action, you can gather details on blocking or wait types being experienced.
Lets assume it's blocking...
The easiest option is to stagger your jobs so that they no longer run in parallel. By eliminating concurrency, you should eliminate blocking.
Another option would be to add ideal covering indexes to speed your deletes. I'm not able to reproduce your issue, but I suspect something along the lines of this might help:
CREATE NONCLUSTERED INDEX MyNewIndex_history_detail
ON dbo.log_shipping_monitor_history_detail
(agent_id,agent_type,log_time_utc);
CREATE NONCLUSTERED INDEX MyNewIndex_error_detail
ON dbo.log_shipping_monitor_history_detail
(agent_id,agent_type,log_time_utc);
