When did you take the class? Could you e-mail the professor and ask him? Or maybe see if he has a website with a syllabus posted?
If “Ulysses” isn’t the poem you’re looking for, I seriously doubt it’s a respresentative work by a famous poet.  My friends are all stumped, and three of them studied poetry as their specialty for their MA. I’ll let you know if anyone gets struck by inspiration.
 My friends are all stumped, and three of them studied poetry as their specialty for their MA. I’ll let you know if anyone gets struck by inspiration.
The only three that come to mind don’t fit the OP criteria in one or more ways, but I’ll mention them anyway:
Rudyard Kipling’s IF, probably the most famous poem in which a father gives advice to a son.  The father’s not a king however (to my knowledge) though he does mention them in the final stanza.
The second is The Bishop Orders His Tomb at St. Praxed’s, in which the title character is (obviously) a bishop rather than a king but a very rich and powerful one, and he’s giving instructions to his (illegitimate) sons in which he mixes in excerpts from his own very corrupt life. It’s also not short, but 125 lines. Link
The real dark horse candidate is Sonatorrek, or (in English) “Lament for my Sons”, by Egill Skallagrimmson (translates as Spearpoint son of Grimskull), a Viking warrior and poet who died more than 1,000 years ago, but I mention it here as it is specifically about sons and it may have appeared in a modern English translation. And of course because I like typing Spearpoint son of Grimskull.
“Odysseus to Telemachus” by Brodsky also kinda fits the description, though it is longer than Tennyson poem posted above.