The Premier League outfit were unable to mark their European debut with a victory in the first leg of their third-qualifying-round tie last week, when they were held to a goalless draw at the Stadion Pod Dubnom in Zilina.
The result, which would have been different had Tom Huddlestone not fired over from point-blank range after seeing a penalty saved, made Hull the only team from Europe's big five leagues to fail to register a victory last week.
However, Steve Bruce's men remain strong favourites to progress at the expense of the Slovakian side ahead of the second leg and will be relishing the opportunity to join the likes of Inter, Tottenham and Villarreal in the play-off round.
Many of the third qualifying round's biggest names are on course to progress from their respective ties.
Italian outfit Torino - making their return to top-level European competition after a 20-year absence - were 3-0 winners when they visited Brommapojkarna of Sweden, so will be satisfied that they have more than one foot in the next stage.
Lyon will be similarly confident when they defend a 4-1 advantage over Mlada Boleslav in France, while Real Sociedad take a 2-0 lead to Aberdeen.
Mainz are by no means assured of their place in the play-off round after only mustering a 1-0 win over Asteras Tripolis of Greece on home soil, and 1978 UEFA Cup champions PSV will defend the same scoreline in Austria against St Polten.
Chikhura's Mikheil Meskhi Stadium will not be at full capacity as they look to edge past Neftci after UEFA handed down a partial closure of the ground following a racist incident in the previous round's first-leg clash with Bursaspor. Chikhura and Neftci are tied at 0-0 following the first leg.
Two teams who were in the group stage of this competition last year face an uphill struggle to repeat that feat this time around, with Chornomorets trailing Split 2-0 and Zulte Waregem 5-2 down against Shakhtyor.
Krasnodar, Omonia Nicosia, Astra, Elfsborg, Club Brugge and Rijeka all appear set to secure a place in the next round after picking up some of the biggest wins in the first-leg matches.
But there is all to play for in eight of Thursday's 29 ties, including the games involving Hull and Chikhura, where the teams could not be separated in their initial clashes.
Elsewhere, Spartak Trnava lead St Johnstone 2-1, Stjarnan hold a 1-0 advantage over Lech Poznan, Atromitos return to Greece 2-1 up against Sarajevo, and Zimbru are in control of their clash with Grodig - leading 2-1.
Young Boys are a goal to the good as they head to Cyprus to face Ermis, CFR Cluj are a goal down against Dinamo Minsk, Rio Ave are 1-0 up against Gothenburg, and Shakhter Karagandy hold a 4-2 advantage over Hajduk Split.
