Letters About Political Issues
The vast majority of letters (about pending issues) written to politicians are read by staff members of the politician (who often code similar responses and reply with a form response, sometimes approved by the politician or a more senior staff member).
Exceptional letters are frequently brought to the member's attention, while boilerplate letters are usually seen by a member only once even if hundreds are received although the number of letter of each type received are typically tallied.
Do we have any way of knowing . . . whether these letters ever have
any influence on the positions that these politicians take?
Politicians use the letters received in a few respects. They gauge how "hot" an issue is to somebody. They are notable when they don't match the politician's expectations about preferences in the district. And, rare, one off personal letters about lower profile issues can provided insights to politicians that may be useful in gaining information about obscure issue of which the politician might otherwise not have been aware.
If an issue is hot, a good politician makes statements about it and votes on it with more attention and care in most cases rather than being flip and making off the cuff arguments that are ill considered.
But, letters rarely cause a politician to change a position on an issue about which the politician has already made a strong ideological commitment unless they are wildly unexpected in volume or in pro/con mix on an issue.
Letters about personal problems in dealings with the government from a constituent of the politician are quickly referred to different staffers who specialize in sorting out bureaucratic tangles that constituents get into, a process called "constituent service."
All of these letters are read, and the politician's constituent service staff (at the state level, some legislatures consolidate most constituent service for the entire legislature, for an entire house of the legislature, or for a political party) addresses issues where the staffer agrees that there is a legitimate grievance.
Sometimes an inquiry will be made with a call from staff or a formal letter signed by the politician, other times the politician will call or there will be a formal letter from multiple politicians who are all experiencing the same problem.
Agencies, in turn, have staff members who are dedicated to responding to these inquiries, which in federal agency vernacular (as my wife who worked briefly in a federal agency related) is called "a Congressional."
If the agency is unresponsive, the issue is sometimes escalated by putting a freeze on some legislative action, like a budget request or a Presidential appointment to the agency, until the issue is addressed, or the issue may be escalated to a superior of the agency causing the problem.
(Most of this answer is based upon personal experience from working in and around legislators in Congress and in state legislatures.)