It is shooting into the light, aperture won't make a difference. F25 will indeed soften the image through defraction but won't affect the flare spots other than to introduce the star effect on the sun.
Some lenses are worse for it than others, and make sure you've not got a UV filter or CPL on the lens (the UV could well introduce flare, a CPL might and in any case shooting direct into the sun it will have a minimal polarising effect) and that your lens is as clean as possible (smears and dirt may worsen flare).
Beyond that, other than not shooting into the sun, you could clone those bits out easily enough. I shoot into the sun a lot and have to make sure my filters (grads, Big Stopper etc) are spotlessly clean and scratch free.