Example: during the encounter phase one investigator was delayed, but during mythos phase he needed to move to other dimension

Q: What to do, to move him or not?

If the investigator is delayed by an Arkham Encounter card and then gets drawn into a gate that opens at their location, they are transported to the other world and are still delayed. Note that when you get hit by two delays like this (one for the Arkham Encounter, one for the gate opening under your feet) you're still just delayed once; there's no "double delay" unless an encounter specifically says, "stay here for two turns," or something similar.

 Delayed means : during your movement phase, if you're delayed, you received 0 movement point and you movement is over. Put your investigator on his feet.

I would say that any event which instruct you to move from your point to another remains valid. Delayed or not.

 Sorry i didnt explain in full

Investigator was half delayed in encouter (encouter said he needs to stay on that location for the next turn), during Mythos phase gate opens on another location and Reconing card (Lurker At The Treshold expansion) said that ivestigator with most power tokens is pulled trough the gate, the delayed investigator has the hiest nuber of power tokens.

So what do I do now?

Well, for starters, the Investigator attempted to remain in place to satisfy the requirement from the Arkham Encounter…I would not penalize the Investigator for not remaining until the next turn.  As to his 'status' ~ he's delayed as he didn't choose to enter the gate as much as he was pulled through the gate.