I do not have access to your data, so bear with me with my example 
First of all:
- you can not want to use
addCircleMarkers(), as these have constant width in pixels (not changing with map zoom - see example below). You need an addCircles() call.
- the default unit for distance in {leaflet} package are meters, you have to multiply your miles figure by 1609.344 (or add dependency to {units}, which seems unnecessary given the task)
Try the code below and watch the difference in behavior between red circles (stable in size, regardless of zoom) an green ones (changing in size with zoom level). Green behaviour is what you want 
library(sf)
library(leaflet)
shape <- st_read(system.file("shape/nc.shp", package="sf")) # included with sf package
cities <- data.frame(name = c("Greensboro", "Wilmington"),
lng = c( -79.819444, -77.912222),
lat = c(36.08, 34.223333))
leaflet() %>%
addProviderTiles(providers$Stamen.Toner) %>%
addPolygons(data = shape, fill = NA) %>%
addCircleMarkers(lng = cities$lng,
lat = cities$lat,
radius = 10,
col = "red") %>%
addCircles(lng = cities$lng,
lat = cities$lat,
radius = 10 * 1609.344, # radius is in meters, adjust accordingly :)
col = "green")