I've done some research on the matter and it seems like there were indeed many Hispanics (Mostly Latin-Americans) who came to the Philippines in the first two centuries of Spanish rule, enough to change the genetics of the ordinary Filipino.
I tracked down some references from the web and these are the numbers which I found so far...
In the initial colonization, there were about 1200 Spanish families settled at Manila and 400 Spanish soldiers guarding them.
So considering that the average count of members per Spanish family were about 6 or 7 in the 1500s, that means that there would be a total of 8400 Spanish living in Manila at the 1500s alone.
Next, we go to Cebu City, at the Visayas, the settlement received a total of 2,100 soldier-settlers from New Spain (Mexico).
Now, if you add Cebu's Mexican population of 2,100 to Manila's population of 8,400 Spaniards, it totals to 10,500 Hispanics. Add to this are the Mexican soldiers stationed at Cavite which were a regiment and would number around 1000...
Adding these should amount to around 11500 Soldiers. To add to this would be the Peruvians settled in Zamboanga City as ordered by Don Sebastían Hurtado de Corcuera.
There is no exact number of them stated, what's only known is that there were a great many of them recruited and considering the transportation of that age, then the colonists would roughly be the same as the figure for the Mexicans settled in Cebu, which is 2,100.
Source:
Then, you would also add the Mexican settlers who settled in the Arrabales of Ermita, which was named because of a Mexican hermit which lived there. The community would count around a thousand Mexican settlers...
Adding them up would produce around 15,600 colonists during the 1500s (Most of which would be Latin-American in origin, Mostly from Mexico and with a tiny contribution from Peru. There would be not much Spanish, there were very few Spaniards who directly migrated to the Philippines then).
In fact, here is a graph which prove that direct migration from Spain was minuscule...

So, during the beginning years of colonization, by the late 1500s, Mexico and Peru sent out a total of 15,600 soldiers and colonists.
15,600 won't seem to be alot to affect the Filipino population, however, we must bear in mind that there were also very few native Filipinos then.
According to a tribute count of all native Filipinos under Spanish governance, the whole population of the Philippines only amounted to about 666,712 people in the islands. Thus, during this era, the Philippines was among the most sparsely populated lands in Asia. In contrast, Japan during that era (the 1500s) already had a population of 8 Million or Mexico had a population of 4 million, which was huge compared to the Philippine's mere 600,000.
Well, I think the 15,600 Latino soldiers and colonists could have adequately affected the 600,000 native Filipinos, in order to spread the European and Native American genes around, in a diluted form, as we see in genetic studies about Filipinos and the 15,600 migrants were only true for those who came during the 1500s, during the 1600s an additional 18,000 people arrived from the Americas, the combined 15,600 migrants in the 1500s and the 18,000 migrants in the 1600s could almost add-up to 40,000 Latino soldiers, way more than enough to affect the 1600s' 700,000 population of native Filipinos.
It seems like my collation for the initial Mexican migration to the Philippines of 15600 in the 1500s, is correct. Since by the 1600s the number is quite close, 18,000.

It's 18000, mostly soldiers. By the 1600s, the population of the Philippines increased to 700,000+, and by that time, the descendants of the 16500 New Spain settlers from the 1500s, began to spread around the population too.
You can look at the immigration records from Mexico, here.
Also, most of the people transported to the Philippines from Mexico were from these locations...
