Limited Express using Hakone Free Pass

I was looking at the Hakone Free Pass, which apparently includes a round trip between Shinjuku station into Hakone. I was wondering if the Odakyu Limited Express Train (travelling between Shinjuku(JR) and Hakone-Yumoto station) is covered fully by the pass? If not, what trains should I take instead? Thank you!

It is, but you cannot use the Romance Car.

Odakyu Limited Express going to/from Shinjuku to/from Hakone is calles Romance Car.

If you want to travel to Hakone only with the Hakone Free Pass, you just take the express, rapid or train other than Limited Express "Romance Car." The Free Pass covers the base fare only. No seat reservation available on those.

So if you want to travel quicker and want to reserve seats, you need to pay extra to get reservations on Romance Car Limited Express.

I was wondering if the Odakyu Limited Express Train (travelling between Shinjuku(JR) and Hakone-Yumoto station) is covered fully by the pass?

"Shinjuku (JR)"?
Odakyu Electric Railway and East Japan Railway Company [JR-East] are not sharing tracks in Shinjuku.
You cannot travel by Odakyu lines from or to Shinjuku Station of JR.

A Hakone Freepass does not cover Limited Express surcharges.
You can make a combination of
a 2-day / 3-day Hakone Freepass (the version covering the Odakyu round-trip basic fares on Shinjuku -- Odawara; 5140 yen / 5640 yen)
and a pair of Romancecar "Hakone" round-trip tickets (Shinjuku -- Odawara or Hakone-Yumoto; 2 x 890 yen).

Romancecar trains take various routes; please make sure to choose Romancecar "Hakone" when you reserve your seat.

The Hakone Tozan Railway section Odawara -- Hakone-Yumoto is inside the free ride area.
(You can travel by bus instead between these two places.)
Romancecar seat reservation cannot be made separately for this section before the boarding date.
You can reserve in advance your seat for the whole route Shinjuku -- Odawara -- Hakone-Yumoto.
http://www.odakyu.jp/english/romancecar/howto/

I was looking at the Hakone Free Pass, which apparently includes a round trip between Shinjuku station into Hakone.

The 2-day / 3-day Hakone Freepass is offered in two versions:
a version to travel only within the free ride area beyond Odawara
(4000 yen / 4500 yen)
and a version which covers the Odakyu round-trip basic fares until Odawara
(5140 yen / 5640 yen: when you travel from Shinjuku).
https://www.japan-guide.com/e/e2358_008.html

The gap in the prices between the two versions is less than the total amount of the Odakyu round-trip basic fares.
So, purchasing the latter version is better cost-wise than paying separately for the former version and for the round trip.

Hakone Free pass is sold by Odakyu railways and is only valid on specific Odakyu railways lines from Shinjuku to Hakone. It is not valid on any JR lines, which are run by a separate company.

The 2-day / 3-day Hakone Freepass is offered in two versions
There are more than two versions, but not many people get the pass activated at Machida (I did last year).

Shinjuku (JR)"?
Odakyu Electric Railway and East Japan Railway Company [JR-East] are not sharing tracks in Shinjuku.
You cannot travel by Odakyu lines from or to Shinjuku Station of JR.

While technically true that Shinjuku JR and Shinjuku Odakyu are two different stations, from a practical perspective they are so close to each other that it feels like one.

I think the OP did already understand that the Hakone free pass is only valid on Odakyu line. And there is an option for the pass to start in Shinjuku or also later, eg Machida or Odawara. Thus should you have an active JR pass you�fd go by JR to Odawara and buy a Hakone free pass only there.

Enjoy your trip to Japan!

While technically true that Shinjuku JR and Shinjuku Odakyu are two different stations, from a practical perspective they are so close to each other that it feels like one.
So close, they even have a transfer gate from one to another.