Moving spatial data off gird cell corners

I have a seemingly simple question that I can’t seem to figure out. I have a large dataset of millions of data points. Each data point represents a single fish with its biological information as well as when and where it was caught. I am running some statistics on these data and have been having issues which I have finally tracked down to some data points having latitude and longitude values that fall exactly on the corners of the grid cells which I am using to bin my data. When these fish with lats and long that fall exactly onto grid cell corners are grouped into their appropriate grid cell, they end up being duplicated 4 times (one for each cell that touches the grid cell corner their lats and long identify).

Needless to say this is bad and I need to force those animals to have lats and long that don’t put them exactly on a grid cell corner. I realize there are probably lots of ways to correct something like this but what I really need is a simply way to identify latitudes and longitudes that have integer values, and then to modify them by a very small amount (randomly adding or subtracting) so as to shift them into a specific cell without creating a bias by shifting them all the same way.

I hope this explanation makes sense. I have included a very simple example in order to provide a workable problem.

fish <- data.frame(fish=1:10, lat=c(25,25,25,25.01,25.2,25.1,25.5,25.7,25,25), 
                   long=c(140,140,140,140.23,140.01,140.44,140.2,140.05,140,140))

In this fish data frame there are 10 fish, each with an associated latitude and longitude. Fish 1, 2, 3, 9, and 10 have integer lat and long values that will place them exactly on the corners of my grid cells. I need some way of shifting just these values by something like plus are minus 0.01.

I can identify which lats or longs are integers easy enough with something like:

fish %>%
  near(as.integer(fish$lat))

But am struggling to find a way to then modify all the integer values by some small amount.