We present the sharpest and deepest near-infrared photometric analysis of the core of R136, a newly formed massive star cluster at the centre of the 30 Doradus star-forming region in the Large Magellanic Cloud. We used the extreme adaptive optics of the SPHERE focal instrument implemented on the ESO Very Large Telescope and operated in its IRDIS imaging mode for the second time with longer exposure time in the H and K filters. Our aim was to (i) increase the number of resolved sources in the core of R136, and (ii) to compare with the first epoch to classify the properties of the detected common sources between the two epochs. Within the field of view (FOV) of 10.8^″ × 12.1^″ ($2.7\,\text {pc}\times 3.0\, \text {pc}$), we detected 1499 sources in both H and K filters, for which 76 per cent of these sources have visual companions closer than 0.2^″. The larger number of detected sources enabled us to better sample the mass function (MF). The MF slopes are estimated at ages of 1, 1.5, and 2 Myr, at different radii, and for different mass ranges. The MF slopes for the mass range of 10-300 M_⊙ are about 0.3 dex steeper than the mass range of 3-300 M_⊙, for the whole FOV and different radii. Comparing the JHK colours of 790 sources common in between the two epochs, 67 per cent of detected sources in the outer region (r > 3^″) are not consistent with evolutionary models at 1-2 Myr and with extinctions similar to the average cluster value, suggesting an origin from ongoing star formation within 30 Doradus, unrelated to R136.