You can use sp_msforeachdb to loop databases:

EXEC sp_MSForEachDB 'USE [?];

print ''Database: ?''
if object_id(''yourtableName'') is not null -- table must exists
BEGIN
   
   -- Do something clever here
   update sometable set somefield = ''somevalue'' where somethingorOther=1

END
'

The procedure runs the provided sql string in all your databases. The database being run is passed in the "?", so one can do things like USE [?] to change to that database.

Note that procedure has some quirks, but works mostly ok, you can search around for alternatives as well.

Inside the string, you check so your table actually exists by doing OBJECT_ID('tablename'). Then, the real problem becomes on how to update your data. I have no idea how you can achieve that if ids are all different, but you can probably use some LIKEs or something like that to match the text.

One complication is that all quotes (') inside the string must be quoted again, this makes query writing a bit cumbersome

First you have to do documentation work.

i )Prepare list of all | ColumnName |TableName | DatabaseName | Server IP| where spelling correction is require

By doing this automatic idea will come. OR you can then update question accordingly here along with sample data.

So far whatever I understood, This is not tested and also ignore syntax error

Customized suiting your requirement. For now it work only in one database.

Create table #Correction(WrongSpelling varchar(200),CorrectSpelling varchar(200))
insert into #Correction(WrongSpelling ,CorrectSpelling ) 
values('Person with a serious metal illness','Person with a serious mental illness')
,('sepprate','separate')

Create table #Dictionary(schema char(3),tablename varchar(100),columnname varchar(100)
, WrongSpelling varchar(200),  CorrectSpelling varchar(200))


create table #temp(schema char(3),tablename varchar(100),columnname varchar(100))

insert into #temp(schema,tablename,columnname)
select table_schema, table_name,column_name 
from information_schema.columns where column_name in('AnswerText','QuestionText')

insert into #Dictionary(schema ,tablename ,columnname , WrongSpelling,  CorrectSpelling)
select schema,tablename,columnname,WrongSpelling ,CorrectSpelling #temp t,#Correction C

Now loop thru #Dictionary using cursor,inside cursor block 
i) use dynamic query for update ii) use sp_executesql iii) use like in predicate.

It is optimize enough,also this is one time query so 
optmization do not matter,accuracy matter.

Test properly

Update A
set QuestionText=CorrectSpelling
from dbo.QuestionnaireQuestionLookup A
inner join #Dictionary dic on a.QuestionText =like '%'+dic.WrongSpelling+'%'

So do the cursor & dynamic query work and let us know