Away from home at the moment, so I can't speak in great detail, but I can speak generally - in brief - to a narrow range of those dates, specifically from the division of the ducal state of Poland as part of the will of Boleslaw III (early 12th century) to reunification of the kingdom under Wadyslaw (spelling?) the Elbow-High in the early 14th.

As it stands, I don't know if I would say the HRE or its emperor had a great deal of "influence" as a single, cohesive actor, apart from allowing and condoning the Teutonic Order's eastward expansion and crusade against the pagan Prussians at the beginning of this period (under the original request of one of the Polish nobles, too - now that's some irony).

Really, in this time frame, the German influence could be divided between two primary camps: 1. the Brandenburg rulers who eventually successfully pulled Silesia away from Polish influence and into the German cultural camp, and 2. the actions of the Order itself, which took over large swaths of Polish territory along the Baltic.

Another pervasive German influence was the continual influence of German settlers who pored into Poland - and much of Eastern Europe - bringing their culture and, arguably more importantly, Magdeburg town laws which removed their towns from the control of local nobility or ecclesiastical authority.

After the death of Boleslaw III, much of the region's influential figures were more focused on regional control and dynastic squabbling than they were in reestablishing a unified Polish Duchy. This continued for around a hundred and fifty years or so.

For reasons I cannot recollect enough to speak to, the attitudes among the Polish rules and various members of the Piast dynasty changed in the mid to late 13th century. I would suspect the effective loss of Silesia to the Brandenburgs, territorial loss along the Baltic to the Teutonic Order, and incursions by the Bohemians(Czechs) (not to mention the Mongolians in 1241-ish) contributed to a feeling of "we're not going to take it anymore." But that's pure speculation.

You could make an argument that the HRE via the Brandenburgs could be seen as partially responsible for getting a non-Pole on the throne: Wencelslaus II of Bohemia was on a tear against the Poles and Hungarians for years and through conquest and political maneuvering, eventually managed to secure the throne as Duke and later King of Poland. However, many of the Polish nobles didn't care for a Bohemian on the crown and after Wencelslaus' and his son's death, the Wladyslaw the Elbow-High managed to be named King. His rule was marked by continued conflict against the German Brandeburgers and the Order. His son Casmir's rule was, as well, but it was also considered to be something of a golden age among the Poles.

Really interesting time period.

Source: Rise of the Polish Monarchy: Piast Poland by Paul Knoll

Polish and German historians have had very different ideas of the origin of Poland, specifically of how important German influence was in its founding and Christianization. In AD 1000, for example, Emperor Otto III seems to have crowned Boleslav I as king of Poland by putting his own imperial crown on Boleslav’s head. In the Polish interpretation, Otto was granting him equal and independent standing; while in the German interpretation, Otto was making him a sort of under-king.

Anyway, in helping to establish Poland and Hungary, Otto III seems to have been trying to show the power of the imperial rank by creating inferior kings around his borders. The Holy Roman Emperors also on various occasions meddled with Polish and Hungarian succession. Those places were still using partible inheritance, so there were always disgruntled claimants to the throne hanging out at the imperial court. So if it seemed politic, the empire might decide to press their guy’s claim against his older brother or whoever the king was at that point.