Translation:Every year, he could only say one word.
That means something else.
"Každý rok směl říct jen jedno slovo." means he only could utter one word and then had to be silent.
"Každý rok směl říkat jen jedno slovo." means he could be talking repeatedly, but he could always only use one single word.
I do not know whether the other can be covered by the English sentence.
Yes, next to tense (past, present, future), Czech verbs exhibit aspect (perfective or imperfective). Most verbs come in pairs like this where one is perfective (říct), where the action is completed, and the other is imperfective (říkat), where the action is ongoing. This is perhaps the most challenging part of Czech grammar.
- Říkal jsem. - I said or I was saying, repeatedly or over a period of time
- Řekl jsem. - I said - and I completed the action of "saying"
- Říkám. - I say or I am saying. Only the imperfective aspect is possible in the present tense, because the action can never be completed (the present is an ongoing moment)
- Budu říkat. - I will say or I will be saying - an ongoing action in the future
- Řeknu. - I will say - once and I'll be done with it. (the present form of the perfective verb ("říct") actually creates the future tense)
These two things don't match up. English has the simple and progressive distinction (such as "I talk" vs. "I am talking" or "I talked" vs. "I was talking") in all tenses. Czech (and most European languages) don't have this distinction at all. Czech has the perfective and imperfective aspect, whereas English doesn't. There is some overlap between these two grammatical phenomena but they just express different things.