You are right, Paco. The word 'problem' is definitely a countable noun. That's why when I first encountered the expression 'have x problem -ing,' perhaps decades ago, I thought it's a rather unusual construction. This peculiarity compelled me to memorize the expression by rote then. Along the way, the eccentricity, if you will, seemed to have been supported by the sentences I occasionally came across. Here is another sentence retrieved from my personal data. If my memory serves me right, the sentence was taken from either a newspaper or a magazine.
Small- and medium-sized businesses have problem getting people to work for them.
If it's a typo, the case is settled.
(p.s.: Remember I always enjoy reading your posts.)
Davkett.
Thank you for your comments. Can't the indefinite article 'a' be an optional item in this particular construction? Well, it could have derived from its analogy, in terms of both construction and meaning, to the expression 'have difficulty -ing' where 'a' is not used.