'have persisting' is correct and usable in written English.
You might use this phrase when discussing someone's ability to remain focused on a particular task. For example: "The team demonstrated their commitment to the project by having persisting focus on the end goal."
Exact(26)
Lyme arthritis must be considered in patients from Europe who have persisting joint effusions.
But even if such treatment is available, some patients have persisting complaints.
Nine of the 12 have persisting amenorrhoea with elevated levels of both gonadotrophins.
Women infected with chlamydia who are enrolled into trials have persisting prevalent infections with an unknown date of infection.
This case illustrates that Lyme arthritis must be taken into account in patients in Europe who have persisting joint effusions.
But even if such treatment is easily available, cheap and safe, some patients have persisting complaints despite contact with the health care system.
Similar(34)
Budget woes have persisted.
Tensions have persisted nonetheless.
And I have persisted.
But troubles have persisted.
But sectarian attacks have persisted.
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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