Grammatically, it's a somewhat challenging passuk. However, the challenging part centers around the phrase אֲשֶׁר יָבֹאוּ, a third of the way through. The beginning is pretty straightforward: "The Nephilim were in the land (or 'on the Earth') in those days and also afterwards..." The grammar here isn't ambiguous. "And also afterwards" can only refer back to the Nephilim. You could end the passuk at that point and have a perfectly sound grammatical sentence.
Also, if you look at the cantillation marks, which are grammatical markers, the main division in the sentence occurs after "bore them." If "and also afterwards" referred to the B'nei Elohim, the division would naturally have occured at that point, but it doesn't.
In terms of narrative, "afterwards" is necessary to foreshadow the post-diluvian existence of Nephilim as reported by the spies at Numbers 13:33 (see Ibn Ezra) and/or the "Anakim" in Joshua 14:15 (see Radak). There is no further discussion of B'nei Elohim which would necessitate an "afterwards," as their only other cameo appearance is in the Book of Job.
As for the second part of your question, it does seem that the Nephilim are being linked to the Giborim (the "Mighty Ones"), since it says "these are" (הֵמָּה), a grammatical tool of referring back to the main subject of a sentence, in this case the Nephilim. This makes sense, since the passuk is best understood as describing how the Nephilim came into existence -- through a union of the B'nei Elohim and the daughters of men -- and then explaining to an audience that had heard of both "Giborim" and "Nephilim" that they are one and the same.
The Torah does this in other places -- for example, in connection with the spies, Numbers 13:3 states, וַיִּשְׁלַ֨ח אֹתָ֥ם מֹשֶׁ֛ה מִמִּדְבַּ֥ר פָּארָ֖ן ... אֲנָשִׁ֔ים רָאשֵׁ֥י בְנֵֽי־יִשְׂרָאֵ֖ל הֵֽמָּה, "And Moses sent them from the Wilderness of Paran ... heads of the Children of Israel they were (הֵמָּה)." The use of הֵמָּה refers back to the previous passuk, a longer stretch than referring to the Nephilim in the beginning of the same passuk here. Similarly, in Numbers 8:16, הֵמָּה refers to the Levites discussed in the previous passuk.
Part of the issue is that the translation you're relying upon is a poor translation, that drastically changes the order of the words and the sentence structure from Hebrew to English. Despite its idiosyncrasies, I prefer the Everett Fox translation (available on Sefaria), which almost always retains the original grammatical structure and often the precise word order. It is not the most "beautiful" translation, but in my opinion it is a great learning tool.