That can actually be correct if you just omit the comma (,) between "rivalry" and "were". That's a kind of sentence where the subject can be found at the end. In detail: In today's rivalry = prepositional phrase were = verb many children = subject In a regular structure, it would be: Many children were in today's rivalry.

There were many children in today's rivalry would be better.

hmm, I'm not sure how to answer the question correctly because it's not too clear to me. There is neither a preposition or an adverb in this example sentence. A possible suggestion: In yesterday's rivalry, there were many children. In today's rivalry, there are many children. "were" is a form of the verb "be" (in case you didn't already know), so we need to have a subject of this verb placed before it in the sentence. Here, "there" represents the situation - Today's rivalry. So to conclude, incorrect: In Manchester are many homeless people. Correct: In Manchester there are many homeless people.

https://dictionary.cambridge.org/dictionary/english/rivalry you need to tell us what the rivalry was over. (a common amongst them thing) keeping the tenses the same. yesterday there was a great rivalry between school children to do well in school. (yesterday implies in history not necessarily yesterday) today there is no rivalry amongst school children to do well in their exams.

the question is nonsense and needs to be rewritten. TODAY'S = present were = Past because your question is ambiguous (lacking context). We can only guess what you wish to say.