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The phrase "approved to run through" is correct and usable in written English.
It can be used in contexts where something has been authorized or permitted to proceed or continue, often in relation to processes or procedures.
Example: "The project has been approved to run through the next quarter, allowing us to meet our deadlines."
Alternatives: "authorized to proceed" or "cleared to continue".
Exact(1)
The targeted three grants, estimated to total about $5 million, had cleared NIH peer review and are underway; one is approved to run through 2012.
Similar(59)
BTeV is a collider experiment approved to run at the Tevatron at Fermilab.
BTeV is a collider experiment that has been approved to run in the Tevatron at Fermilab.
Therefore, only 161 Islamic theologians were approved to run for the assembly of experts.
A total of 17 states (so far) have been approved to run their own exchanges.
It just needed to run through him.
It is scheduled to run through May 9.
The production is to run through Sunday.
Nair planning to run through all details.
Production is expected to run through August.
Do not try to run through crowds.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com