Mark records that activities and descriptions of the Holy Spirit in a similar manner to the way that Matthew described the Spirit. (See “The Holy Spirit in Matthew.”)
Thus, in Mark, as in Matthew, we see John the Baptist proclaiming that Jesus will baptize with/in the Holy Spirit (Mark 1:8). We also see the Holy Spirit descending on Jesus after John baptized him (Mark 1:10) and leading (driving) Jesus into the wilderness for a time of fasting and temptation (Mark 1:12).
Mark also warns against blaspheming the Holy Spirit (Mark 3:29). Mark also records that Jesus said that the Spirit inspired David to pen one of the Psalms (Mark 12:36). Finally, in Mark, Jesus tells his followers not to be concerned about what they should say when they are arrested because the Holy Spirit will speak for them (Mark 13:11).
All of these passages concerning the Holy Spirit are recorded by both Mark and Matthew. There are several additional passages in Matthew that Mark does not record. However, in the passages that Mark does record, he occasionally gives more (or different) information.
For example, in Matthew, the Spirit leads Jesus into the wilderness, but in Mark, the Spirit drives Jesus into the wilderness. In Matthew, John the Baptist says that Jesus will baptize with the Holy Spirit and fire, while Mark simple has John saying that Jesus will baptize with the Holy Spirit.
While some may be concerned about these differences, I am not. To me they show the perspectives of two different authors.
But, the fact is, there is very little (if any) new information about the Holy Spirit in Mark compared to the information that we found in Matthew.