Contracts created by means of eMClient do not appear on cell phone, on the contraty contacts created by means of cell phone always appear in eMClient. Can you fix it?
Check when you are creating a contact in eM Client that you have selected the correct folder to save it in.
If it is Local Folders/Contacts/ that means it is stored only on your computer so will not sync with Google. By clicking on the folder you can select your Google account contact folder.
In case I was unable to sync locally stored contacts I would have posted a message saying that I was unable to sync local contacts 
Google contacts sync doesn’t work, clear?
Actually, you said that contacts are syncing with Google, just not contacts you create in eM Client. This is a typical situation where new contacts are saved to the wrong folder so do not sync.
But if syncing with Google was not working, then how are you seeing those contacts added by your phone?
Can you please look in Menu > Tools > Operations, either in Errors but probably in the Log and see what error is there regarding your Google Account.
It may also be that the contact cache is somehow corrupted. Right-click on the contact folder in eM Client and choose Properties > Repair. See if that helps.
There have also recently been problems (which the EMC team is aware of and working on) specifically with the Google Calendar/Contact sync hitting the daily API transaction limit and erroring out once it does. @burivoy if that is at least in part what you are experiencing you should see it clearly in your logs as Gary mentioned.
Let us put in this way: google contacts sync is unstable, only half of the contacts created in eMClient appear on cell phone, but 100% of contacts created in cell pnone appear in eMClient. Contact sync in unreliable.
What category do the contacts in eM Client have that are not syncing with your phone?
What category do the contacts in eM Client have that are are syncing from your phone?
My Contacts
Both the contacts from your phone and the new eM Client contacts have the My Contacts category?
Listen, contacts created in cell phone do not have any category, there is no such field at all, they are “just Google” contracts.
That is what I asked, and you said they had My Contacts category.
Open one of the eM Client contacts that are not syncing with your phone and remove the category from it. See if that shows on the phone because it looks like your phone is not syncing the sub-folders (categories) in Google Contacts.
And eMClient doesn’t allow to create/have contacts without any category at all? Great idea. This walkaround doesn’t work.
If you want the contact to have no category, save it in the topmost folder.
What you see as folders are just Google labels (categories).
Open your web browser and go to your Google Contacts. Open the My Contacts folder. You should see all the contacts there from eM Client that are not syncing with your phone. Still in the web browser, create a new contact in that folder and see if it syncs with your phone.
TruePhone dialer and contacts manager has Category field.
But what happened when you tried the test I suggested?
Nothing. I wrote that that walkaround did not help. Switching to alternate contacts manager TruePhone which supports categories solved the issue somehow. As soon as gmail webinterface proved that contacts sync between eMClient and server works it was not eMClient issue; it was a problem of syncing between gmail server and cell phone.
Meanwhile, what about yandex imap folders encoding in UTF7 and google calendar timezone issues?