bügeln
to iron
To smooth clothing or other textiles with a hot iron in order to remove wrinkles.
Ich muss noch meine Hemden für die nächste Woche bügeln.
I still have to iron my shirts for next week.

to iron / to smooth over
Loading your saved state.
to iron
To smooth clothing or other textiles with a hot iron in order to remove wrinkles.
Ich muss noch meine Hemden für die nächste Woche bügeln.
I still have to iron my shirts for next week.

The verb 'bügeln' (to iron) is closely related to the nouns 'das Bügeleisen' (the iron, the device for smoothing) and 'der Bügel' (e.g. a clothes hanger). The act of ironing creates smooth surfaces and clean edges.
Kannst du mir zeigen, wie man das Bügeleisen richtig benutzt?
Can you show me how to use the iron properly?
In its literal meaning, 'bügeln' (to iron) is often combined with the textiles that are being smoothed. Typical examples are 'Wäsche bügeln' (to iron laundry), 'ein Hemd bügeln' (to iron a shirt), or 'eine Bluse bügeln' (to iron a blouse).
Am Sonntagabend bügle ich immer meine ganze Wäsche für die Woche.
On Sunday evenings, I always iron all my laundry for the week.
In the figurative meaning 'to correct a mistake,' 'ausbügeln' (to smooth over/fix) is a separable verb. The prefix 'aus-' stands at the end in a main clause.
Mein Kollege hat den Kunden verärgert, und ich musste seinen Fehler wieder ausbügeln.
My colleague upset the customer, and I had to smooth over his mistake again.
In the figurative meaning, 'ausbügeln' (to smooth over/fix) refers to abstract problems. Common combinations are 'einen Fehler ausbügeln' (to fix a mistake), 'ein Missverständnis ausbügeln' (to clear up a misunderstanding), or 'ein Problem ausbügeln' (to fix a problem).
Sie versuchte, das peinliche Missverständnis schnell wieder auszubügeln.
She tried to quickly smooth over the embarrassing misunderstanding.
to correct
'korrigieren' and 'ausbügeln' can both mean putting a mistake right again. 'korrigieren' is neutral and works well for mistakes in texts, numbers, statements, or plans. 'ausbügeln' is more figurative and more colloquial. It often means eliminating a problem or the consequences of a mistake. So the difference is: with 'korrigieren', the mistake itself is the focus; with 'ausbügeln', it is more about the damage or the unpleasant situation afterward.

Nach dem Missverständnis musste Mehmet den Fehler beim Kunden ausbügeln.
After the misunderstanding, Mehmet had to smooth over the mistake with the customer.
Die Verkäuferin korrigiert den Preis auf der Rechnung.
The saleswoman corrects the price on the invoice.