This isn't an area I know a lot about, so my comments are going to be largely drawn from Peter Jackson's The Mongols and the West, 1221-1410. But the short answer is: not good.

The Mongols of the Ilkhanate flirted with Christianity for some time (and one Ilkhan, Öljeitü, was even born a Christian—though he later converted) and were viewed as a potential ally by Catholic Europe against the Muslims after their 1260 defeat at Ayn Jalut signaled the end of any threat they might have posed to the West. (I very briefly describe Crusader-Mongol—which largely reduces to Crusader-Ilkhan—diplomacy in this answer to another question a few months ago.)

The same was not true of the khans of the Golden Horde,* who continued to be viewed as hereditary enemies of Christendom—first as pagans, and then as Muslims. Part of this might be that unlike the Ilkhanate, Golden-Horde-occupied territory was (a) conquered from Christian princes rather than Muslim ones; and (b) was close enough to Catholic territories to still pose a direct military threat. The Teutonic Knights' interactions with the Golden Horde—which we may regard as a specific arena of Papal relations with the Golden Horde—were therefore largely military ones designed to foil any Mongol attempts to invade further European territories. This largely antagonistic relationship persisted despite the fact that both the Teutonic Knights and the Mongols struggled with a common foe for much of the period of the Golden Horde's establishment—the Lithuanians.

[* note: the term "Golden Horde" is not accepted without reservations by all modern Mongolists, as it's a 16th-century Russian imposition upon the past. For most of the period we're concerned with here, there were actually two hordes, a White and a Blue. (The "Golden Horde" refers to the combination of the two.) What you're asking about here is the relations of the Teutonic Knights with the White Horde, the more westerly of the two. But I've referred to it as the "Golden Horde" throughout my answer for the sake of convenience.]

All that being said, our knowledge of Teutonic Order-Mongol [from here on, "Mongol" will be used to refer to the Golden Horde exclusively] interactions is pretty spotty, as the Golden Horde largely acted either alongside or through proxies like the Russian princes or the Lithuanians. It can therefore be difficult to figure out what engagements the Mongols were involved in, if there were no Mongols present on the field. But here are a few of the interactions that Jackson lists, which should be sufficient to give an idea of the tenor of Teutonic-Golden Horde relations.

• A Mongol attack on Prussia (part of an invasion of Poland launched in 1259) in 1259-60 inflicted heavy losses on the Order, and in August 1260 Pope Alexander IV asked the Order to assist the Polish crown if another major attack were launched. (This didn't end up happening, as the Order instead had to devote its forces to quelling domestic unrest in 1261-62.) [Jackson, 123]

• Rumors of engagements between the Teutonic Knights and the Mongols in 1260 (part of the aforementioned campaign; a combined Teutonic-Polish army is said to have beaten the Mongol army but this is probably apocryphal) and 1263. [132, 201]

• In 1286 the Knights evacuated four Prussian strongholds based on news of an approaching Mongol army that never materialized (probably inspired by preparations to invade Poland again). [206]

The next century and a half is rather obscure, for the reasons stated above; the Golden Horde by this time had largely withdrawn from direct involvement in European affairs. Jackson notes another facet of Teutonic-Mongol relations in this period, though, namely client states' interactions with both powers. Though not an example of direct relations between the knights and the Mongols, we have plenty of examples of minor rulers who "played the field" by engaging in diplomatic relations with the Order while in a tributary relationship with the Golden Horde. Jackson gives special attention to the cases of the Ruthenian princes Lev II and Andrei (both died in 1322/3) and their successor Yuri II (died 1340), who both made pledges of friendship to the Teutonic Order. [209-210]

We should not let such pledges lead us into thinking that the Order and the Golden Horde had reached some sort of rapprochement. In another episode narrated by Jackson, Casimir III of Poland tries to secure the Order's support for a campaign against the Lithuanians by mentioning that he had also enlisted the assistance of seven Mongol princes. The knights not only do not help Casimir (they had already made their peace with the Lithuanians, at least temporarily), but they also tell him off to the Pope, who gives the King a stern rebuke for his behavior. [211]

Casimir's statement also signals a shift in the Golden Horde's relationship with the Lithuanians, who had initially been Mongol clients but by the mid-fourteenth century were engaged in outright warfare with the khan's armies. By the early 15th century, then-Grand Duke Vytautas had become something of a kingmaker with respect to the fragmentary and chaotic remnants of the Golden Horde. As part of this dynamic, Vytautas seems to have enlisted a Mongol contingent into the Lithuanian forces at the Battle of Grunwald/Tannenberg (1410) against the Teutonic knights. [219] This is the last engagement I am aware of between the Knights and any representatives of the former Golden Horde.

Jackson, Peter. The Mongols and the West, 1221-1410. Harlow, England ; New York: Pearson Longman, 2005.

EDIT: In his notes, Jackson draws on the following article, which I haven't had a chance to read but may address your question more completely:

Sarnowsky, Jürgen, ‘The Teutonic Order confronts Mongols and Turks’, in Malcolm Barber, ed., The Military Orders. Fighting for the Faith and Caring for the Sick (Aldershot, 1994), pp. 253–62.

Not so good, but could be worse. The Teutonic Order was setup in the region partially to assist the Christian populations of ‘the East.’ Though this may imply all Christians it was mainly Catholic Hungary and Poland. One of the main focuses was Prussia; however, in the Mongol invasion by Batu Khan, the Teutonic Knights were to assist Henry the Pious of Poland against the Mongols at the Battle of Legnica (1241) where Bajdar, Orda and Kedan will crush the Christian army before their reinforcements could arrive. By the end of 1241, the Great Khan Ogedei will pass away, this forces Batu to return to the Kurultai for elections where the Mongols will run into a political stand-off and eventually at the death of the Great Khan Mongke (1259) the Mongol Empire will fall into a Civil War that lasts for 4 years.

We now have to examine the political situation of the Eastern portion of Europe. Though the Pope and the Catholic leaders had their issues with the Orthodox peoples, it was now overshadowed by the fact that a large pagan power was now coexisting with them. Things seemed shaky for a couple of reasons; firstly, the German leadership was not simply going to unite for Holy War. Secondly, Poland was heavily divided amongst its duchies and last but not least, Hungary who had taken many Cuman refugees in during and after the Mongol invasion of the region, now had a considerable pagan population that even played a role in their politics as Queen Elizabeth (Daughter in-law of King Bela IV of Hungary) was a Cuman herself. The Teutonic Order was well aware of the dangerous situation that was posed not only by the Golden Horde, but the fact that the Lithuanians and Prussians being pagan themselves, were going to ally themselves with the Mongols as they had been under great pressure from the Churches; and as a matter of fact, the Mongols will launch raids into Poland with the assistance of the Lithuanians. This prompted the Teutonic Order to encourage King Vacalav III of Bohemia to attempt to unite the Polish Duchies under his rule. This ironically caused more division amongst the Europeans and allowed for the Mongols to spy and raid the Eastern people freely. The Popes were having no shortage of anxiety here as they feared that the Eastern European Catholics were going to fall under Pagan rule as the Rus did, especially when Bohemia and Hungary were going at it. And perhaps it may have eventually; however, the Golden Horde was at war with the other Mongol power, the Ilkhanate.

Though the Poles and the Teutonic Knights seemed to face imminent pressure from the Prussians, Lithuanians and now the Mongols, there was no more than small skirmishes between the two, and is in fact more likely that the skirmishes were with the Rus that were just being slightly backed up by Mongol forces. Many European nobles eventually tried to use different diplomatic ways of keeping themselves safe from the Golden Horde, may it be paying them tribute or assisting them against a common enemy like Italian traders in the Black Sea for example, and eventually, beginning with the period known as the Dual KHANship, supporting opposing Mongol factions. Ultimately, it seems that the Teutonic Knights were just worried that eventually Mongol carnage would come to effect their position in the region, but with the division of the Golden Horde, their fears never came to light.

The Mongols and the West Peter Jackson