what precautions should be taken to minimise errors due to changes in
pH and blood gas values in the blood  sample from te moment the blood
exits the finger-prick wound until the sample is presented for
analysis?

Is this a trick question? pH and blood gas values are not normally
evaluated from "finger prick wounds".
The distal ends of the fingertips would not yield an arterialized
blood sample which would reflect an accurate picture of the acid-base
balance of the body.  Heel-sticks on neonatal patients are done
(usually) when an arterialized sample cannot be readily obtained from
typical sites (radial artery in the wrist, brachial artery in the
crook of the elbow, or umbilical artery in neonates with a catheter in
that site) and even then the heel stick values after "arterializing"
the site with a warm compress, only reflect a value relative to that
particular patient's peripheral circulatory status.

If you want to know what precautions should be taken to minimize
errors ... pH/blood gas...arterialized heel stick, then I would
suggest 1) eliminate the presence of any air bubbles in the sample
during collection 2) ice the sample 3) perform an immediate analysis
4) mix the sample in the pipette with a metallic "flea" and magnet
prior to analysis.