I don't see any answers or additional questions, so I think that my topic was hard to understand or not interesting for community. So, I will share solution that I'm implemented.
In ES 8 I found recommendation that there should be single instance for ES client in the application. Later I found that such recommendation presents in ES 7 that we're using now. I haven't seen this recommendation before.
After moving ES client to be a singleton, I see this results. All requests to ES runs less than 25ms (instead of the first one after startup). Value closer to 25ms could be found in processing of ES queries with reported value of "took" close to 9ms.
It looks much better than previous situation when query with reported "took" value equal to 2ms was processed up to 60ms. Now this queries could be processed even faster than 15ms.
We think that there are no way to improve speed of processing for now. I hope that this decision (use ES client as singleton) will help someone to improve his speed of running queries.