I agree with PJT.
Yet, there is a way to insert the verb "poder" with "es posible que": "Ayer, mientras iba manejando, me puse a pensar que es posible que nunca pueda haber una cura contra el coronavirus."
This sentence questions the capability to ever produce a cure for the coronavirus, as humans seem to be always outsmarted by the bug. -> There's a possibility that there can never be a cure for the coronavirus.
"Ayer, mientras manejaba, me puse a pensar que es posible que nunca haya una cura contra el coronavirus."/ "Ayer, cuando iba manejando, se me ocurrió que podría nunca haber una cura contra el coronavirus."
Your original sentence considers any bunch of reasons why the cure may never exist, but it doesn't rule out the capabilities humans have to actually develop one in a remote future anyway. -> It's possible that there never may be a cure.
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