You may want to tune your images, or perhaps use the width and height arguments of IMG tag more actively.
Consider this piece of code; note the use of single quotes to delimit strings, so that double quotes will remain safely inside (and utilized by the HTML call). Also note how I replaced the zero at the end of your link with a 1; a neat trick I found at https://support.exclaimer.com/hc/en-gb/articles/360018586691-How-to-host-an-image-using-Dropbox
library(sf)
library(dplyr)
library(leaflet)
a_point <- nominatimlite::geo_lite_sf("Clermont-Ferrand") %>% # looks about right
mutate(label = paste0("lo & behold: a hedgehog",
'<img src=',
'"https://www.dropbox.com/s/9njtb9o9lql25o0/atelerix_algirus.jpg?dl=1"',
'alt="no longer stolen from wikipedia"',
'width="300"',
# 'height="250"',
'>'))
leaflet(data = a_point) %>%
addProviderTiles("Esri.WorldImagery") %>%
addMarkers(popup = ~label) %>%
setView(lat = 50.073658, 14.418540, zoom = 4) # view centered on Prague (just because...)
