The problem is more likely to be gummed tappets than a blocked pick-up. The pick-up is a coarse strainer, only really catching larger bits, leaving the filter to collect finer particles. The top end has smaller oilways, so more susceptible to contaminants.
Oil grade is more important than where you bought it. Tesco will source from a reasonable source, but if you've used 20/50 instead of 10/40, the thicker oil will not go through the oilways properly until hot.
Get some flushing oil and change the oil for this. Run it as directed, then change for the correct grade (check the handbook, but probably 10/40) with a filter change. If lucky this will sort it. If not, you may be into new tappets, or removal for thorough cleaning.
Moral: Good oil, and regular changes are expensive, but cheaper than engine repairs.