
absegnen
to approve
Colloquial for giving official agreement or approval for a plan or undertaking.
Der Vorstand muss das neue Budget nur noch absegnen, dann können wir starten.
The board just has to approve the new budget, and then we can start.
to approve / bless
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to approve
Colloquial for giving official agreement or approval for a plan or undertaking.
Der Vorstand muss das neue Budget nur noch absegnen, dann können wir starten.
The board just has to approve the new budget, and then we can start.

to bless
In a religious context, to give a blessing, often at the end of a ceremony or for consecration.
Der Priester wird das neue Gemeindehaus feierlich absegnen.
The priest will solemnly bless the new parish hall.
'Absegnen' is a separable verb. The prefix 'ab-' is placed at the end of the sentence in a main clause. In a subordinate clause or infinitive with 'zu', the verb stays together.
Meine Chefin segnet den Urlaubsantrag hoffentlich ab.
Hopefully my boss will approve the vacation request.
The word consists of the prefix 'ab-' and the verb 'segnen' (to bless). The prefix 'ab-' gives the word the meaning of 'abschließen' (to complete) or 'fertigstellen' (to finish) here. So one gives one's 'Segen' (blessing) in order to bring a matter to a close.
Sobald die Pläne abgesegnet sind, beginnen die Bauarbeiten.
As soon as the plans have been approved, construction work will begin.
The meaning 'genehmigen' (to approve) is colloquial. In formal or official texts, this word is generally avoided.
Kannst du kurz meine E-Mail absegnen, bevor ich sie verschicke?
Can you quickly approve my email before I send it?
Whether the colloquial or the religious meaning is intended is easy to recognize from context. In the office, in politics, or in project planning, 'absegnen' always means 'genehmigen' (to approve). Only in a church context is the religious blessing meant.
Der Stadtrat hat den neuen Bebauungsplan abgesegnet.
The city council approved the new development plan.
to approve
Genehmigen and absegnen can both express official approval. Genehmigen is the formal, often administrative term. Absegnen means the same thing here, but sounds colloquial and is therefore more suited to an informal conversation.

Im Gespräch sagt Paul, die Chefin habe den neuen Projektplan abgesegnet.
In the conversation, Paul says that the manager approved the new project plan.
Im offiziellen Schreiben genehmigt die Chefin den neuen Projektplan.
In the official letter, the manager approves the new project plan.