No it wouldn't. Once the man had made clear that he wouldn't financially support the child, the woman could decide - in full possession of the facts - whether to continue with the pregnancy
Just because termination is legal doesn't mean that it is easily accessible, particularly through the NHS, right up to the legal deadline.
A woman can only legally have a termination up to 24 weeks, and in actual fact it can be very difficult to obtain one after 12 weeks and nigh on impossible in many areas (except in exceptional circumstances) after 16 weeks.
It usually also takes at least week (often more like a couple of weeks) to actually arrange a termination - if you go through the NHS rather than a private/charity clinic it can often take 3 weeks to book.
So most women would need to have made the decision to terminate a pregnancy by the time they were about 10 weeks pregnant. After that their options may be incredibly limited, and in many cases for many women may be non-existent.
So littlemissrain at what point would the man need to declare he was "financially aborting"? It's all very well to say Obviously, a man should only be able to financially abort before the baby is born but the idea that a man has the choice to abort up to 40 weeks, whereas for most women there is only a choice up to 10 weeks is clearly unfair. If the man gets the right to abort right up to birth (or even up to say 13 weeks in most cases) where does that leave the pregnant woman who up until that point may have felt she had the full support of the father and is now left with no choice to terminate?
You also say this should apply as long as he was made aware of its existence at that time - what if he isn't? Can he financially abort an existing child? What about the cases where women do not know they are pregnant in time to get a termination (as I said before in many areas you would need to know you were pregnant and have made the decision to terminate by 10 weeks in order to get a termination through the NHS)?