Forest NINJA (please sign all posts with at least a first name plus an initial. It is a rule here so we can address each other in a polite manner. You can add your name to your profile),
I have explained this several times here on NMB, but again:
TAMAHAGANE is a very basic form of metallic iron. It can have differnt amounts of impurities (= NOT ALLOY METALS !). The low temperature of its production process (around 1250 - 1300°C) is not high enough to reduce other alloying element oxides that may be present in the iron ore (there are always other elements like manganese, chromium, silica, copper, titanium, a.s.o.).
You cannot "mix in" other metals into the process unless you throw modern alloyed steel in the TATARA. Another method would be the OROSHIGANE process where iron can be carbonized but also modern alloyed steel could be introduced. Traditionally, this is of course not done. Steel alloys can only be made in hot liquified state - roughly 1.600°C or more.
The TATARA process does not produce melted (= cast) iron, at least not in considerable amounts but more as an unwanted by-product. Nevertheless, even these small amounts of cast iron are sold and used by some swordsmiths to increase the carbon content of their steel.
The TATARA method (comparable to the early European bloomery furnace) produces a very pure iron as far as alloy metals are concerned, but it is not "clean" in hindsight to impurities like silica and metals that are attached to it like titanium. So even in the 'best' TAMAHAGANE, you will find very small amounts of impurities.
The traditional swordsmith does not care for them; he is solely interested in the carbon content. Carbon is introduced into the iron by the charcoal used in the process, but is is not automatically distributed homogeneously. There are spots with higher heat in the TATARA furnace where the iron takes on more carbon than in "cooler" places. This is why repeated forging (= stretching out, folding and fire-welding) is required later for refining.
From a metallurgical view of the subject, differences in the appearance of sword-steel and its properties are mainly due to the work of the swordsmith. How he treated the metal in the long process has probably much more influence than tiny amounts of a few impurities. However, it is known that some elements like titanium can have an effect even in very small amounts in an alloy.
Since there are no actual comparative scientific tests on authentic Japanese sword blades, we will probably not know much about this subject. However, ASANO TARO, a swordsmith in GIFU, has executed final fracture tests on a blade (on YouTube under asanokajiya).