MMarco

The issue was to trigger the automatic citizenship. In terms of PP, OP can apply at the embassy back in his country when the kids finish the round trip. So, he doesn't need to apply for the PP in US.

Neither will the State department when it rejected the son’s passport application 
Original Poster. You. 
 
And with that, I’m out.

https://www.uscis.gov/policy-manual/volume-12-part-h-chapter-4
 
 
Legal custody refers to the responsibility for and authority over a child. For purposes of this provision, USCIS presumes that a U.S. citizen parent has legal custody of a child and recognizes that the parent has lawful authority over the child, absent evidence to the contrary, in all of the following scenarios:[8]

…
 
A biological child born out of wedlock who has been legitimated and currently resides with the parent;
does not apply to OP
 
A child of divorced or legally separated parents where a court of law or other appropriate government entity has awarded primary care, control, and maintenance of the child to a parent under the laws of the state or country of residence.
does apply to OP.  
 
You need to focus on this if you want your 17 year old  son to be a citizen before he turns age 18.  Just trying stuff and hoping it will work can result in:
 
* citizenship being denied 
 
* citizenship being erroneously granted and then years later, State Department or USCIS revokes citizenship.  Revocation doesn’t require due process in court though your son(s) can try to sue to over turn it.  

yes, 
the kids mother and myself were married at the time we have the kids, and after divorce we never been in court for legal custody,
but as she "the kids mother "was busy she ask me to take care of the kids,
so I send the kids to my mother, and they living with my mother for over 6 years now   
 

Just between you two. Try and work something. 
 
No requires document to avoid losing citizenship. They just come in and it's automatic. However, they have to be under 18 when they pass the point of entry check in. I'll suggest not to wait to have them come in last minute because things happen. Latest I'll suggest is a week before he turns 18.

My opinion? Make a round trip on a weekend and take advantage of the automatic citizenship that he can get if he enters under 18 years and save $$ and USCIS headaches down the road. Trust me, if you have an opportunity like this to strike many USCIS birds with same stone, take it. 
 
When he comes, he gets a stamp on PP that serves as a temporary GC for 1 year. He can leave and go back to take care of his studies. The benefits of this are as I explained above.

Assuming you have physical and legal custody of him, he should enter the USA before he turns age 18 so that he is automatically a USA citizen.  If he waits until after age 18, he can file N-400 in 2027.
 
File N-600 after he successfully gets a U.S. passport and U.S. passport card.  You get just one chance at an N-600.  

I'm so depressed, forget about me just be ready for your test and hope your interview be easy all the best

no thing....

so happy for you and you family Congratulations 

I applied by mail (hard copy), did you applied online ?
I don't think that missing part will cause a delay, but I recommend to tell the Officer once you enter your intv   

thanks for your support and beaning positive

both I751&N400

You can go to DHS website and request Ombudsman to review you case. It worked for @LETSGOROCKETS so you may want to try that. 
 
At this point the backlogs are severe and current government doesn't have incentives to process N400 faster for coting purposes. So, I am assuming that these delays are going to linger for a while.

JUST NOW ADDING 8 MOOOOOOOOOOOOOOOORE MONTHS the PROCESSING TIMES ......