Include Columns in Index : SQL Server
Actually if you get Actual Plan, you will be able to see, indexes that modified by every update.

When you do update, Cluster Index and all indexes that keys and included column , had updated, will be modified.

Example 1 :
update #Temp set ClientName = 'foo' 
Execution plan : The clustered index will be update

Example 2:
update #Temp set Balance = 'foo'
Execution plan : The clustered index and [IX_Temp_RegistrationNumber] will be update.

Tip :If your table haven't so many indexes or update operations isn't high, then there isn't any performance issue.