Can taking Finasteride and Avodart at the same time affect the performance of the Avodart i.e. will the drugs compete?

They won't counteract, but I suggest seeing a hair loss expert now to evaluate and go over your condition and provide you information about your best treatment options. There are great non-invasive options like prp/progesterone and/or hair transplantation. See an expert. Best, Dr. Emer

Finasteride is 60% effective at blocking DHT but has a half life of only 6 hours.  Dutasteride at the dose you are taking is 90% effective.  the other difference is that Finasteride works on only 1 of the 3 DHT receptors whereas Dutasteride works on all 3.  Its like getting an A on your test vs a D-.  I know which I would choose.  I wouldnt recommend taking both together though.
The other thing that neither of these medicines in pill form is the ability to block the DHT which converts at the level of the scalp and binds to the hair shaft there.  They also do not include a hair growth stimulant. For best hair growth you should also be including those items in your treatment too.
See a hair regrowth expert who has a fully comprehensive hair regrowth program that effectively stops hair loss AND regrows your own Hair.

Avodart blocks up to 90% of DHT, while finasteride blocks up to 70%.  I see no benefit in taking both on a daily basis.  I have patients using finasteride daily, whose hair loss has continued to slowly progress,  for whom I suggest adding Avodart 2x per week, mostly because it is so much more expensive, and because its biological 1/2-life is longer, so the effect may persist at less than a daily dose.  There are many causes of shedding such as stress, stopping rogaine, or sometimes seasonal shedding.  You might see a Dermatologist if scalp symptoms persist.

No, taking them together will only enhance the overall dihydrotestosterone (DHT) blockade. It will not counteract.

Both dutasteride and finasteride are competitive inhibitors of 5 alpha reductase. There is no point to take both of them at the same time