die Verbesserung
improvement
A change that makes something better than before; progress.
Das neue Update bringt viele Verbesserungen für die Software.
The new update brings many improvements to the software.

improvement / correction
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improvement
A change that makes something better than before; progress.
Das neue Update bringt viele Verbesserungen für die Software.
The new update brings many improvements to the software.

'Die Verbesserung' (the improvement/correction) is the noun form of the verb 'verbessern' (to improve/correct) and is derived from the adjective 'besser' (better). It describes the process or result when something is made better.
Wir arbeiten ständig an der Verbesserung unserer Produkte.
We are constantly working on improving our products.
To express what is being improved, German usually uses the genitive or a prepositional phrase with 'von'. The genitive is more common in written language.
Die Verbesserung der Luftqualität ist ein wichtiges Ziel.
Improving air quality is an important goal.
To describe the extent of an improvement, adjectives such as 'deutlich' (clear/significant), 'erheblich' (considerable), 'spürbar' (noticeable), or 'geringfügig' (minor) are often used.
Seit dem Training sehe ich eine deutliche Verbesserung meiner Leistung.
Since the training, I have seen a clear improvement in my performance.
The meaning 'Korrektur' (correction) is almost always used in the context of texts, homework, or exams. Signal words are 'Fehler' (mistake), 'Aufsatz' (essay), 'Klausur' (written exam), or 'Lehrer' (teacher).
Bitte schreibt alle Verbesserungen aus der Klassenarbeit in euer Heft.
Please write all the corrections from the class test in your notebook.
expansion / removal
'der Ausbau' and 'die Verbesserung' can both mean that something gets better. 'der Ausbau' fits especially when something existing is made larger, stronger, or more efficient, for example a network, a service, or skills. 'die Verbesserung' is more general and emphasizes only the result: something is better afterward than it was before. When it is about greater scope or more capacity, 'Ausbau' sounds more natural. When it is about quality, performance, or progress in general, 'Verbesserung' fits better.

Nach dem Training merkt Paul eine deutliche Verbesserung seiner Kondition.
After the training, Paul notices a clear improvement in his fitness.
Der Ausbau des Radwegs beginnt nächste Woche.
The expansion of the bike path begins next week.