I'll be going to UK as sponsored visitor - am I allowed to apply for Multiple visa entry to UK?
All UK visitor visas (or to be exact, 'entry clearances') are multi-entry for their duration. This holds for all visitors whether the visa is 'sponsored visitor' or not. The rule governing this is explicitly spelled out in Paragraphs 20A and 25A of the rules. Their rationale is that if you are going to overstay, you will do it after arriving and not travel around to foreign destinations first.
There are other classes of entry clearances that are one-time only, but these are in the Tiered category.
When you receive your passport, you can double check to see an imprint on the upper left: "MULT". Expect the same 'grilling' on each and every control point. What that means is that you are required to have a landing interview upon each entry into the UK (regardless of whether you already had one earlier).
The landing interview will be conducted by an Immigration Officer and he/she will focus primarily on Paragraphs 40 and 41 of the rules. Because you are Filipino (which is a visa-national country), he/she may also probe Paragraphs 11 and 24, and most importantly Paragraphs 30A, 30B, and 30C. So carry all your evidence with you for each landing interview. I have provided the specific paragraphs so that you can read them before you travel.
Update 2 August 2015
Owing to a rule change in April 2015, references to Paragraphs 40 and 41 above are obsolete. Those rules were changed and consolidated into Appendix V.
The entire set of UK immigration rules is at https://www.gov.uk/government/collections/immigration-rules