mehr
more
Expresses a larger quantity or a higher degree of something; comparative of 'viel' (much/a lot).
Ich hätte gerne mehr Kaffee, bitte.
I’d like more coffee, please.

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more
Expresses a larger quantity or a higher degree of something; comparative of 'viel' (much/a lot).
Ich hätte gerne mehr Kaffee, bitte.
I’d like more coffee, please.

'Mehr' is invariable. It does not take endings, regardless of whether it comes before a masculine, feminine, or neuter noun or which case follows. It always remains 'mehr'.
Ich brauche mehr Zeit.
I need more time.
'Mehr' is the comparative form of 'viel' (much/a lot). It is used to compare quantities.
Sie verdient mehr Geld als ich.
She earns more money than I do.
The combination 'nicht mehr' is a fixed unit and means 'no longer' or 'over'. It expresses that a state or action has ended.
Seit seinem Umzug sehe ich ihn nicht mehr.
Since he moved away, I don’t see him anymore.
'Mehr' is part of many fixed expressions. 'Immer mehr' means a continuous increase, and 'mehr oder weniger' means 'approximately' or 'more or less'.
Immer mehr Menschen ziehen in die Stadt.
More and more people are moving to the city.
much / a lot
'viel' and 'mehr' both talk about quantities or a degree. 'viel' simply describes a large amount: 'viel Arbeit' (a lot of work), 'viel Zeit' (a lot of time), 'viel Wasser' (a lot of water). 'mehr' is the comparative form of this and means: a larger amount than before, than needed, or than someone else has. The words can be confused because both have to do with quantity. If you just want to say that the amount is large, 'viel' fits. If you are comparing or mean something additional, you need 'mehr'.

Am Morgen brauche ich mehr Zeit, wenn der Bus zu spät kommt.
In the morning, I need more time when the bus is late.
Bei schlechtem Wetter verbringen wir viel Zeit zu Hause.
In bad weather, we spend a lot of time at home.