An assembly is provided for a turbine engine. This assembly includes a combustor, a fuel-air mixer and a mixer guide. The combustor includes a bulkhead. The fuel-air mixer includes an inner passage, a sidewall and a fuel circuit. The inner passage extends axially along an axis within the fuel-air mixer. The sidewall extends circumferentially around and axially along the inner passage. The fuel circuit includes a first fuel passage and a first fuel nozzle outlet fluidly coupled with the first fuel passage. The first fuel passage is embedded within the sidewall and extends along the inner passage. The fuel circuit is configured to direct fuel into the inner passage through the first fuel nozzle outlet. The mixer guide couples the fuel-air mixer to the bulkhead. The mixer guide is configured to slide axially along the fuel-air mixer.