The Greek word used in Mark 8:11-13 for "sighed deeply" means an intense grieving, inaudible prayer, or groan.  This is intercession of the Holy Spirit, with groanings that cannot be uttered.  Some Spirit-filled Christians believe this means groanings that cannot be uttered in normal speech and that this refers to speaking in tongues. 

This actually refers to a type of intercession different than speaking in tongues.

In John 11:33 and 38, Jesus groaned in the Spirit twice, and in those cases there were no words uttered.  It was exactly as the bible said, a groaning in the Spirit.  Everyone who has the indwelling presence of the Holy Spirit has or will have this happen to them.  

Paul was referring to this in Galatians 4:19 when he spoke of travailing in birth for the Galatians.  This groaning of the Holy Spirit is not just a grief but a groan of anger and resistance against Satan in our lives. 

Many times, Christians don't discern this because they think it is just their own grief over the situation.  But, this is the Holy Spirit desiring to intercede with us against our problems.

Jesus drew on this ministry of the Holy Spirit.  John 11:33 and 38 says, that Jesus groaned in the Spirit twice when He raised Lazarus from the dead.  What frailty did Jesus have that He needed this ministry of the Holy Spirit?  Jesus had no sin, but He did have an frailty.  It was His physical mind.

Even a sinless human mind could not comprehend raising a man from the grave after four days.  If Jesus needed the Holy Spirit to help Him when He didn't know how to pray, then this should certainly be an important ministry of the Holy Spirit in our lives.