
die Mühe
effort
The effort or expenditure of physical or mental energy required to complete a difficult task.
Die ganze Mühe hat sich am Ende gelohnt.
All the effort was worth it in the end.
effort / trouble
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effort
The effort or expenditure of physical or mental energy required to complete a difficult task.
Die ganze Mühe hat sich am Ende gelohnt.
All the effort was worth it in the end.

to take the trouble
To take on the effort of doing something, often as a sign of care or to make someone happy.
Er hat sich die Mühe gemacht, für alle Gäste persönlich zu kochen.
He took the trouble to cook personally for all the guests.

to have difficulty
To have difficulty doing or achieving something.
Ich habe große Mühe, mich bei dem Lärm zu konzentrieren.
I have great difficulty concentrating with all the noise.
'Die Mühe' (effort/trouble) belongs to a word family with the adjectives 'mühsam' (laborious/arduous) and 'mühevoll' (painstaking/laborious). All of these words describe an activity that requires a lot of effort and energy.
Der Aufstieg auf den Berg war sehr mühsam.
The climb up the mountain was very arduous.
'Mühe' (effort/trouble) is often used in fixed expressions. Very common ones are 'sich Mühe geben' (to make an effort), 'sich die Mühe lohnen' (the effort was worth it), and to manage something 'ohne Mühe' (without effort/easily).
Wenn du dir in der Schule Mühe gibst, wirst du gute Noten bekommen.
If you make an effort at school, you will get good grades.
The expression 'sich die Mühe machen' (to take the trouble) is reflexive and is often expanded with an infinitive construction with 'zu'. The person making the effort does it for themselves, so 'sich' is in the dative.
Sie hat sich die Mühe gemacht, einen Kuchen für mich zu backen.
She took the trouble to bake a cake for me.
The expression 'Mühe haben' (to have difficulty) is often used with an infinitive construction with 'zu' or with the preposition 'mit' + dative. It expresses that an action is difficult.
Ältere Menschen haben oft Mühe, die moderne Technik zu verstehen.
Older people often have difficulty understanding modern technology.
work
Arbeit and Mühe can both refer to the effort required for a difficult activity. Arbeit focuses more on the task or the objective amount of work, while Mühe highlights the effort someone personally experiences. Use Mühe when you want to emphasize how much effort someone had to put into something.

Für den Bericht musste ich mir viel Mühe geben.
I had to put a lot of effort into the report.
Für die Arbeit an dem Bericht brauchte ich drei Stunden.
It took me three hours to work on the report.