I wish to draw an important distinction regarding your question. In general, can prophets foresee the future? Sure they can - if God reveals it to them. Let's review what prophecy is.
Hear now My words:
If there is a prophet among you,
I, the Lord, make Myself known to him in a vision;
I speak to him in a dream.
So prophecy is revelation from God, given through visions or dreams. It is definitely NOT a "power" that people have, or are given. That is actually what makes prophecy different from witchcraft or sorcery. Sorcery is the use or acquisition of power through magical means. Power that one then uses of his own initiative. Notice this even in the words of Simon the Sorcerer to Peter and John in Acts 8:19, for which Peter rebuked him strongly:
“Give me this power also, that anyone on whom I lay hands may receive the Holy Spirit.”
So my answer is NO, neither Bob Jones nor Paul Cain had the "power to see the future". If they did, they would not be prophets, but sorcerers. In the jargon of the Charismatic movement, people refer to the "anointing" of a prophet. While any believer (and even occasionally non-believers) can receive visions and dreams from God, some have a more potent 'anointing' and have more frequent and more intense experiences.
There is (or was - some channels on YouTube where their stuff was posted have recently disappeared) ample documentary material available online of visions and dreams that both Bob Jones and Paul Cain claimed to have received from God. Some of those are verifiable / falsifiable predictions of specific future events. Others are future oriented but not predictive: "When you see X, respond in this way." Others are personal prophecy, meaningful only to the individuals concerned. Others are revelations of wisdom or teaching and not predictive in nature. All of these are considered valid forms of prophecy by Pentecostal and Charismatic believers.
The point being, even though most of their prophecy is not verifiable / falsifiable, does not make it junk. It is a matter of whether you trust them as legitimate prophets. Those claims of theirs that are verifiable, absolutely should be verified, as a means of establishing trust or identifying false prophets.
I recall the specific prediction from Bob Jones in the question, about the Kansas City Royals winning the World Series. This prediction was not given just to show off an ability to predict the future. God is not a sports bookie, and it would be akin to tempting God to expect a prophecy to be specific enough that it could be used for betting or profit.
The point of that prophecy was to provide a 'sign'. The victory of the Kansas City Royals was to be a sign of a spiritual victory of some kind in the area of Kansas City that would come to have historical importance into the future.
“It looks like Kansas City is down and out and will never make it and suddenly at 5 till 12 the Lord brings the victory in. That’s the spiritual message over this movement breaking forth in this city. When it looks like it is five till twelve and Kansas City is counted out. Suddenly at 5 till 12 the Lord’s breaking in victory.”
While phrased as a prediction, it wasn't really, in my opinion. If the Royals had lost the World Series that year, they would have looked to a future year for that sign (and the Royals won again in 2015), not concluded that the prophecy was false.
As to the verifiability that the claim was made in advance of the Royals' victory - it's going to be tough. There was no such thing as YouTube or Instagram or TikTok in 1985. Unless somebody had a VHS camcorder and somehow documented and published it in advance of the game, there will probably be no proof. Maybe, just maybe, if the prediction was made far enough in advance, somebody might have an old pamphlet or transcription of it with a date printed on it. (If you're willing to believe it's not a more recent forgery.) Otherwise you'll just have to take Mike Bickle's word for it.
One of the most explosive and verifiable prophetic claims of recent history comes to us through Times Square Church pastor David Wilkerson, who died in a car crash in 2011 but on March 7, 2009 posted on his blog this very specific and dire prediction. At the time, he was openly doubted even among Pentecostals, but his prediction was completely fulfilled starting in the summer months of 2020.
Major cities all across America will experience riots and blazing fires... There will be riots and fires in cities worldwide. There will be looting—including Times Square, New York City.
Very, very few people in 2009 believed that in the near future, societal disruption of that scale would be possible in America. But now, it cannot be denied that his prophetic prediction was true and accurate.