GNSS antenna upgrade

They are certainly quite good for base applications. Unaided GPS you might see 15-20% shrink in CEP. Size and mass not helpful in rover applications, or ones where signals are obscured/blocked.
Sort of rings shown in your post can be machined from 65mm Aluminium stock. Looking for trench depths >1/4 wavelength. 
Many use fairly generic antenna elements, one could certainly mount an ANN-MB on the top plate, or other bulkhead antenna. I have some old L1 AeroAntenna models that I plan to retro-fit as some point with multi-band antennas. Those looked to be a cast/machined construction, with a 1″-14 marine thread to pole mount.
This is one I had machined for stock, with 5/8″-11 thread tripod mount. https://portal.u-blox.com/s/contentdocument/0692p000006YSrqAAG
I’d recommend trying a 14-16″ (350-400mm) diameter aluminium plate ground plane and see if that improves your situation. Also establish a local base rather than use one 30-35 KM away.

Hello, everyone! Thank you a lot, ardusimple community for you advices! Thanks to them, I got a coordinate error in the plan of 3 cm, at a height of 25 cm in the forest in RTK. Settings of my receiver: elevation mask 0 degrees, all frequencies and satellite systems are included, SBAS systems, measurement frequency 1 Hz, Internet LTE 1/5 division. The antenna is mounted on a steel plate with a diameter of 12 cm, above the antenna is a protective cover made of ABS plastic with a thickness of 4 mm. The base station is 35 km away. Type of messages from her: rtcm 3.2. For control, I carried out measurements at a geodetic point in the forest, the result was compared in the local coordinate system. Observation conditions: 0-70 degrees of elevation mask was covered by pine trees in snow with sky gaps. In the spring, in warm weather, I plan to conduct an experiment: measure a number of points with a total station and a receiver in RTK in the forest, on the border with the forest and near the forest. At the same time, the receiver should be under the trees. I plan to tabulate the results and calculate errors in the plan and height. I think it will be interesting for Ardusimple comunity