How to shift a CRS by an observed offset

I have a drone image is which is geo-referenced over a Google Sat. image. These align well, and both use WGS84-Pseudo Mercator (EPSG:3857).

I have a set of data points in a shape file which are WGS-84 (EPSG:4326); and QGIS is set for on-the-fly re-projection. Overlaying the datapoints over the images, I notice that all the data points are offset from the underlying physical objects seen on the images -- for the most part the shift is consistent across the entire data set of 20 or so points (allowing for GPS errors at the time of data collection, I suppose)

My questions is this:

How does one physically shift the data set, that is by shifting the CRS to a custom CRS, by that small offset. I would assume that I create a custom CRS, based on the existing WGS84, and then add some terms to the command line which accomplish the shift in the X and Y directions, by the 10-20 feet. However, I can't find an instruction tool/set online that explains this. Help....?

Alternatively, I would have hoped for a way to merely enter a vector to define that shift (that is, to create a vector from a data point to the what should be the underlying physical object in the drone photo)....but I don't see that capacity in the software.

Yet another alternative is that there is some odd/weird CRS in the data collection which resolves the shift, and/or a weirdness in the WGS-84 pseudo-mercator projection around where my data lives on the ground. So, I wonder if there is a way to discover which other formally named projections in QGIS live within 10feet of my datapoints (and therefore by using that existing alternate, I get my data points shifted).

I am using QGIS 2.6.1. I realize I could drag each individual data point, manually. Part learning curve, with this question, in advance of a task that will have far more points.