From the tutorial for the ESP8266 (I don’t have one of these) it looks like you only have two options – timer and reset.
 
Using this sentence “The RST pin of the ESP8266 is always HIGH while the ESP8266 is running. However, when the RST pin receives a LOW signal, it restarts the microcontroller.” you will need to provide a low signal to the RST pin to get the 8266 to restart. Checking the data sheet for your sensor will tell you what it does when triggered. If it shorts then connect it between ground and the RST pin.You may need to create a short latching circuit for reliability.
 
Using an ESP32 gives you many more options to wake from deep sleep.

So from what I can see it sends out a high signal 3 v so I am thinking that I would have to put that signal through a N/O set of contacts. When motion is seen it would send its signal to GPIO 14(D5)  to do the reset from RST to GND . Just wondering if I did that, then I probably don’t need any code to accomplish it provided I send the signal long enough to fire that relay . 

No code necessary. Once the sensor fires and it takes RST pin low, the microcontroller restarts.

I have the NodeMcu ESP8266  running and I have turned down the sensitivity somewhat , lower the delay setting to about half (approx 2 1/2 minutes . I left it running  in a room unoccupied and got detections on my telegram 8:17,8:20, 8:24,8:28and 8:31. Thought i may be the cat but looking at the times I suspect that it is just going off for no reason .Any thoughts on this? Pretty much given up on trying to get this thing to work in deep sleep mode also.

I don’t know what sensor you have but, if it’s one that is super cheap and comes with a kit, then it’s probably gonna fire for no apparent reason (I know my one does).

 Steve   
 HC-SR501 is the sensor I have, not very expensive but it does have LHI 781 sensor. I did change the settings for sensitivity and adjust the settings for time delay to a lower setting. 
     What I found was the sensor puts out a high signal as an out put, actually tied a led to it to watch it and set it for the length of on time, 5 seconds. 
     It still isn’t stable enough to put it somewhere as an intrusion alarm 🚨but is a little better. Testing it more tonight. 
     Wondering if I use a reset to put it into deep sleep if it would be a bit more stable.Any idea as how it could be wired using a relay or preferably a NPN transistor. I have an idea but not sure of it. If I was to take the output of the sensor to the base of the transistor then send low to rst ???

Yes. You would have to send a low signal to the ESP8266 RST pin.
Regards,
Sara

This article shows you how to use a transistor as a switch.
 
Everyone out there has had trouble with false triggering of the HC-SR501 sensor. This video will give you tips on making it better.

     Thanks Steve, interesting that in the video ,the op used 3.3v which is what I already had but then  moved the jumper on the 3 pin jumper to retrigger rather than single and then  took the 3.3v supply to pin 3 on the  jumper . So I guess I am trying that solution to see if that stabilizes the motion sensor , also extending the connections from the ESP8266 so it will move the sensor further away from the ESP8266. Not a fan of that because I am trying to keep a small footprint.
    As far as the transistor , i have used a transistor as a switch on a previous project but wasn’t really clear as to how I would connect it to this HC-SR501.

I’m thinking that if you want a stable motion sensor you should be looking at a different one than you have. The constraints are just not worth the effort required.
 
The article I linked should have taught you how to use the transistor as a switch. Can you tell me why you are having issues in your specific situation?

     Just having trouble wrapping my head around it , I am not good at written instructions, I am more of a hands on person. That’s why I have issues here , reading some of this stuff is not really sinking in , so I struggle with a lot of it. 
     Concepts is another thing, I really have a desire to learn this stuff but then I don’t have the imagination to create my own. 
     So no , I don’t think you can really help me much further, I just have to work harder at it and it will sink in eventually. Thanks 

     Just having trouble wrapping my head around it , I am not good at written instructions, I am more of a hands on person. That’s why I have issues here , reading some of this stuff is not really sinking in , so I struggle with a lot of it. 
     Concepts is another thing, I really have a desire to learn this stuff but then I don’t have the imagination to create my own. 
     So no , I don’t think you can really help me much further, I just have to work harder at it and it will sink in eventually. Thanks 

Yes, time is the thing needed. It took me a while when I first started out with all the different types of transistors and what they could do. I wondered why you even needed them when a NAND gate could do pretty much anything you could want 😉 When I was starting out, all you could use were books. Good luck in your further endeavors.