In his book On the Trinity, Augustine (354-430 AD) writes the following regarding Mark 13:32:
Augustine - On the Trinity, Book 1, Chapter 12
Again, "Of that day and that hour knows no man, no, not the angels which are in heaven; neither the Son, but the Father". For He is ignorant of this, as making others ignorant; that is, in that He did not so know as at that time to show His disciples: as it was said to Abraham, "Now I know that you fear God", that is, now I have caused you to know it; because he himself, being tried in that temptation, became known to himself. For He was certainly going to tell this same thing to His disciples at the fitting time; speaking of which yet future as if past, He says, "Henceforth I call you not servants, but friends; for the servant knows not what his Lord does: but I have called you friends; for all things that I have heard of my Father I have made known unto you"; which He had not yet done, but spoke as though He had already done it, because He certainly would do it. For He says to the disciples themselves, "I have yet many things to say unto you; but you cannot bear them now". Among which is to be understood also, "Of the day and hour". For the apostle also says, "I determined not to know anything among you, save Jesus Christ, and Him crucified"; because he was speaking to those who were not able to receive higher things concerning the Godhead of Christ.
Above, Augustine states that Jesus actually did know the day and hour of His return. To support this, he first appeals to Genesis 22:12, in which the angel of the Lord, speaking for God, says, "now I know that you fear God", which Augustine apparently interprets to mean something like, "I declare to you that I know that you fear God". He then mentions that Jesus planned to tell His disciples this information "at the fitting time", despite the fact that nowhere in Scripture is there any record of any of the disciples ever being told this information, and actually, in the passage in question, He instead speaks as though they will not know until it is very near, and so should keep vigilant on that basis (Mark 13:18-19, 13:29, 13:33-37). Finally, Augustine appeals to 1 Corinthians 2:2, in which the Apostle Paul states, "For I determined not to know anything among you, save Jesus Christ, and him crucified". Here, "know" basically means "declare". Therefore, Augustine is arguing that this is what Jesus meant when He said that He did not "know" - He meant He would not declare it to them, although He did know it.
First, it should be stated that a word having more than one meaning does not mean that any meaning for the word can be swapped out at will, regardless of context. Even if the word "know" had 5 meanings or more, in this context, it still clearly means "have knowledge of":
Mark 13:32-33
32 But of that day and that hour knoweth no man, no, not the angels which are in heaven, neither the Son, but the Father.
33 Take ye heed, watch and pray: for ye know not when the time is.
Above, Jesus uses the same instance of the word "know" for Himself, the angels, and people in general. Unless it means "have knowledge of, and also declare" when applied to each, then making an exception to where it means that for Jesus is acontextual. It does not mean, "have knowledge of, and also declare" when applied to men and angels. It means "have knowledge of", and He groups Himself in with them in saying that like them, He does not have that knowledge. This is also supported by verse 33, in which "know" definitely means "have knowledge of".
So, the word "know" having other meanings in other contexts is irrelevant, and making this argument is an attempt to read ambiguity into a passage in which the word "know" is not contextually ambiguous at all. The Apostle Paul in 1 Corinthians 2:2 was using what is basically an idiom, and it was clear what he meant. Jesus in Mark 13 is not doing that.
Additionally, in the account in Matthew, Jesus specifically says that "the Father alone" knows the day and the hour. So, in addition to explicitly saying that He does not know it, He also explicitly affirms that "the Father alone" knows it. If this meant "have knowledge of, and declare", then it would be saying that only the Father would declare it. But, Augustine says that later, Jesus was going to tell His disciples this information, which would mean for the Father, "know" would have to have some other meaning from even that.
In summary, in order to avoid the conclusion that Jesus lacked information of when He would return, Augustine equivocates on the meaning of the word "know". But, examining the context of the passage in question demonstrates that the meaning of "know" is not ambiguous, and here, it means "have knowledge of", with Jesus confessing that He did not have it.