In the UK, a collective can be treated as a plural. While in the US, it's generally considered a singular.
But, that aside, the two components of the sentence should be in agreement.
So it would be one or the other of the following:
The government are being flayed for their abysmal performance.
The government is being flayed for its abysmal performance.
Which of those two is specifically correct would be a matter of style and regional use.
Although they and their can be used as a gender-neutral singular third-person pronoun, that only applies to people (or creatures or certain things that have been treated in a gendered way). Object and concepts don't have a gender.
In short, when discussing individuals and the pronoun used to describe them, consider the following (discounting gender specificity):
✔ No one was allowed to see their marks.
✔ No one was allowed to see one's marks.
✔ No one was allowed to see his marks.
✔ No one was allowed to see her marks.
✘ No one was allowed to see its marks.
But the reverse is true of objects or concepts that have no gender:
✘ No car was allowed to park with their doors open.
✘ No car was allowed to park with one's doors open.
✘ No car was allowed to park with his doors open.
✘ No car was allowed to park with her doors open.
✔ No car was allowed to park with its doors open.