Hello! The answer to this question is "It's complicated..." If I understand what you are asking, it is useful to compare Earth with Mercury.

The solar wind is a stream of charged particles or plasma that mostly consists of electrons, protons, and alpha particles. The solar wind streams outward from the Sun, carrying with it an interplanetary magnetic field that emerges from the Sun. The interplanetary magnetic field is embedded within the solar wind solar wind plasma and fills the entire heliosphere.

When the solar wind reaches the vicinity of Earth, it directly interacts with Earth's magnetosphere, which is the extension of Earth's magnetic field out into space. On the day side of Earth's magnetosphere, a shock wave that scientists call the bow shock, forms at a distance of about 10-12 Earth radii (64,000 to 76,000 km) away from Earth. This bow shock is similar to the bow wave that forms in front of a boat moving through water. Just earthward of the bow shock, is a turbulent region called the magnetosheath where the solar wind flow and interplanetary magnetic field is diverted around the Earth's magnetosphere. At the earthward edge of the magnetosheath, at a distance of around 9-10 Earth radii (57,000 to 64,000 km) is a boundary called the magnetopause. At the magnetopause, a process called magnetic reconnection can occur. Magnetic reconnection is not well-understood and is the topic of one of the satellite missions I work on, NASA's Magnetospheric Multiscale (MMS) mission. The basic idea of magnetic reconnection is that if the interplanetary magnetic field is oriented in the correct direction relative to the Earth's magnetic field at the magnetopause, they can merge which allows solar wind plasma to enter the magnetosphere. Magnetic reconnection is also related convective processes that help circulate plasma inside the magnetosphere. The processes of magnetic reconnection and convection within the magnetosphere all take place at distances of thousands of km from the Earth's surface.

Thus, the solar wind doesn't generally interact directly with the Earth's atmosphere, so it seems unlikely that the solar wind would directly interact with the geomagnetic dynamo below the Earth's surface. The main effect of the solar wind on Earth's magnetosphere is to produce magnetic reconnection and convection that circulates plasma inside the magnetosphere. Stronger than normal solar wind associated with coronal holes on the Sun or coronal mass ejections (CMEs) can compress the magnetosphere, speeding up these processes, and creating a geomagnetic storm. The strong electrical currents generated by changing magnetic fields and charged particles moving in Earth's magnetosphere during a geomagnetic storm can become diverted into the ionosphere, producing auroral displays. These electric currents in Earth's magnetosphere can also reach down to the surface causing geomagnetically induced currents (GIC) in electrical power systems, which I've heard was the topic of a recent National Geographic article. However, I do not know of any studies showing that the magnetospheric current systems during storms affect the geomagnetic dynamo far below Earth's surface.

However, the situation on Mercury may be different. We don't understand a lot about Mercury's magnetosphere because our data on its magnetic field come from only the Mariner 10 flyby in 1975 and the NASA MErcury Surface, Space ENvironment, GEochemistry, and Ranging (MESSENGER) which orbited Mercury from 2011-2015. Mercury is the only other rocky planet to have a magnetosphere and internal magnetic field thought to be generated by a dynamo below the planet's surface. However, although Mercury is believed to have an iron core that is at least partially molten, Mercury's surface magnetic field strength is only about 1% that of the Earth's. We don't yet understand why Mercury has such a weak magnetic field. Dynamo models for Mercury have been proposed that suggest the thickness of the liquid outer core or the precipitation of solid iron play a role. There are also theories of alternative dynamo mechanisms that produce weak fields by invoking feedback between the solar wind, magnetospheric, and core dynamo fields. When considering these mechanisms and whether or not something similar could happen inside Earth's interior, it is important to remember that Mercury is very different from Earth. Mercury's magnetosphere is much, much smaller than Earth's magnetosphere. This due to a combination of Mercury's small size, weak magnetic field, and lack of a dense atmosphere and ionsophere. The interactions between Mercury's magnetosphere and the solar wind are similar to those at Earth in some ways, and very different in others due to the size of Mercury's magnetosphere and the fact that it is much closer to the Sun. The key factors that influence the dynamics of Mercury and Earth's magnetospheres and their internal structures are very different in many ways.

The European Space Agency (ESA) and the Japan Aerospace Exploration Agency (JAXA) launched the BepiColombo mission to the planet Mercury in 2018 and it is expected to arrive at Mercury in December 2025. Hopefully this mission will help us better understand Mercury's magnetosphere and how its internal magnetic field is generated. And maybe what we learn will help us understand our own planet too.